Integral Calculus for Beginners

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INTEGRAL

CALCULUS

FOR

BEGINNERS

CALCULUS

INTEGRAL FOR

WITH

AN

BEGINNERS

INTRODUCTION

THE

TO

DIFFERENTIAL

STUDY

EQUATIONS

BY

JOSEPH FORMERLY

FELLOW

M.A.

EDWAEDS, OF

SIDNEY

SUSSEX

MACMILLAN AND

COLLEGE,

AND NEW

YORK

1896 All

rightsreserved

CAMBRIDGE

CO.

OF

First

Reprinted With

and

additions

GLASGOW

:

BY

PRINTED

ROBERT

Edition, 1891,

1890.

corrections,

AT

MACLEHOSE

1893.

1892, 1894

THE

;

reprinted

UNIVERSITY

AND

1896.

PRESS

CO.

PREFACE.

THE

introduction

to

for

suitable

a

a

it does

therefore

the

of

omission

It

will

be

of

all

found,

and

calculation

of

solids

of

afforded the

to

Centroid, As in want

of

it

or

seems

Applied of

the

in

and

Some of

such

of

undesirable

the

path

with

the

M298720

be methods

also

method

position of

should

is

applications

Moment

that

Mathematics

acquaintance

a

the

surfaces

general

the

obtaining

value

also

as

and

the

cesses pro-

Quadrature,

useful

as

which

reading.

indication

other

rather

ordinary

volumes

the

student

employed

later

a

the

Rectification

but

subject

fully treated,

are

Integral Calculus,

be

to

for

that

its

for Beginners,

the

left

however,

revolution.

the

of

Like

subject.

completeness,

at

best

of

methods

the

the

portions

as

integration

principal

of

aim

sound

a

Integral Calculus,

the

beginning

usually regarded

are

of

form

to

Calculus 'Differential!

the

at

intended

study

student

companion, not

is

volume

present

of

a

Inertia. of

a

blocked of

student

by

a

solving

vi

PREFACE.

elementaryDifferential Equations,and time

that

his

of

study and

exhaustive

added

of

the

includingall with

meet

Linear

such

kinds

his

reading

consistent The been

subjecthas with

the scope

carefullymade of

the student the

"

treated

attackingthe

of the

and originality

A

able consider-

be

explained

harder

ingenuity,though

in

to

been

actuallyset

which

usuallyindicated.

I

am

in

the

sets

at

generallyof

are

character, and

worked

call for few

a

greater

present any

A largeproportionof difficulty.

examples have sources

somewhat

have

text

firmly fixed

chapters.These

miscellaneous

considerable

be

is

illustrate the

to

examples should

may

cients, Coeffi-

present work.

selected

"

general

fullyas

as

that the several methods

so

book-work

before ends

these

to and

Constant

with

of the

or

of the

elementaryparts of

they immediatelyfollow.

number

mind

been

been

likelyto

is

examples scattered throughoutthe

articles which

the

solution

solution of the

Equation

tematic sys-

Analytical Statics,

the

to the

a

has

account

student

of

same

complete

some

of

the

as

Particle,and

a

Differential the

more

brief

ordinarymethods

Rigid Dynamics. Up

the

subject in

the

treatise,a

in

of

Dynamics

in

stopped for

elementaryforms occurring, leadingup

more

by

be

should

course

at the

these

examinations, and

indebted

for them

are

PREFACE.

acknowledgments

My

of

works

the

of

Treatises

of, Bertrand

Greenhill's

which

Calculus, consult

with

My

thanks

kindly the

of

many

sent

great

me

desirable

1894.

the

especially

to

the

Chapter

the

advanced

more

to

on

Todhunter,

and

to

the

on

fessor Pro-

Integral

student

may

advantage.

and

to

friends

several

suggestions

plan

of

the

JOSEPH

October,

degree

some

writers

interesting

valuable

scope

more

and

due

are

modern

the

but

in

due

are

treated

subjects

vii

who

with

regard

work.

EDWARDS.

have

to

CONTENTS.

INTEGRAL

CALCULUS.

CHAPTER

NOTATION,

I.

SUMMATION,

APPLICATIONS. PAGES

Determination

of

1

Area,

an

3 "

......

Integration Volume

of

from

the

"

13

10

Revolution,

"

CHAPTER

GENERAL

Fundamental

9

4

Definition,

II.

FORMS.

STANDARD

METHOD.

19

14

Theorem,

"

.......

Nomenclature

and

21

20

Notation,

"

......

General

obeyed

Laws

by

the

Integrating

22

Symbol, .

Integration Table

of

of

xn,

23"26

x~l,

26"28

Results,

CHAPTER

METHOD

Method The

of

Changing

Hyperbolic

.

the

III.

OF

Variable,

SUBSTITUTION.

29

32 "

33

Functions,

36 "

......

Additional

Standard

Results,

37"41

CONTENTS.

IV.

CHAPTER INTEGRATION

PARTS.

BY

PAGES

Geometrical

of

Parts"

Integration"by

Product,

a

....

48"49

Proof, of the

Extension

50"52

Rule,

V.

CHAPTER

FRACTIONS.

PARTIAL Standard

55

Cases,

57

"

........

Fraction

General

43"47

nominator, De-

and

Numerator

Rational

with

58"61

VI.

CHAPTER

Integrationof

METHODS.

STANDARD

SUNDRY

f^L

65"68 .

...

J \/K of Sines and

Products

Powers

and

Powers

of Secants

Powers

of

or

Tangents

+ o

cos

77

,

79"83

x

CHAPTER

VII.

REDUCTION

Integrationof xm-lXP, Reduction

Formulae

Reduction

Formulae

where

for

for

X

Evaluation

of

FORMULAE.

=

a

+

bxn,

/ xm~lXpdx,

/ sii\nxdx,

.

.

.

87"89 90

93

"

....

/ sin^a; cos^a: dx,

.

.

.

.

.

94

"

95

IT

7T

j"z

78

"

.....

etc.,

"

76

"

.....

Cotangents,

or

69"74

.

.

.

75

Cosecants,

/rfv a

Cosines,

r -i

I

ain^x cos^x

dx,

.

96

"

102

CONTENTS.

xi

CHAPTER

VIII.

MISCELLANEOUS

METHODS. PAGES

/^).fx.9 X. Y

Integrationof

109"117

.......

J

i\f

Integrationof some SpecialFractional Forms, General and Geometrical Illustrations, Propositions Some Elementary Definite Integrals, Differentiation under an IntegralSign, .

118

119

"

.

120

"

124

.

125

....

128

....

"

"

127 129

IX.

CHAPTER RECTIFICATION. Rules

for

Formulae

Curve-Tracing,

of

for

Equation,

of Pedal

Illustrative

Closed Curve,

a

Evolute,

an

Intrinsic Arc

and

for Rectification

Modification Arc

135"137

.......

Examples,

.

13S

"

140

.....

143

........

144

........

Curve,

139

"

149 150

........

CHAPTER

X.

QUADRATURE. Cartesian

Formula,

Sectorial Areas. Area

of

Other Area

a

Curve,

a

of

'.''-. .

two

of Curvature

Radii

the

168"175

........

176

.......

CHAPTER

of

164"165

166"167

CorrespondingAreas,

Volumes

and

.........

Pedals,

SURFACES

157

161"163

.......

.....

Curve,

"

158"160

.......

Expressions, Evolute,

Areas

Polars,

Closed

between

153

........

AND

VOLUMES

Revolution, Surfaces of Revolution,

"

177

XI. OF

SOLIDS

OF

REVOLUTION.

.......

183"184

.......

185"187

CONTENTS.

xii

PAGES

Theorems

of

Revolution

Pappus,

of

a

188

.......

Sectorial

192

Area,

......

XII.

CHAPTER ELEMENTS

SECOND-ORDER

191

AREA.

OF

MISCELLANEOUS

APPLICATIONS. Surface

Cartesian Integrals, of

; Moments

Centroids

Element,

Inertia,

195

.

199

.....

Polar Element, Integrals, Centroids,etc.,Polar Formulae,

204

.....

DIFFERENTIAL

of

a

Differential

207

"

EQUATIONS. XIII.

CHAPTER

Genesis

201

"

202"203

Surface

EQUATIONS

198

"

OF

Equation,

ORDER.

FIRST

THE

211"214

.....

215

Variables Linear

Separable, Equations,

216

XIV.

CHAPTER

EQUATIONS

OF

ORDER

FIRST

THE

(Continued}. 221

Homogeneous Equations, One

Letter

Clairaut's

226

"

227"229

Absent,

230"233

Form,

XV.

CHAPTER

EQUATIONS

219

"

OF

THE

EXACT

ORDER.

SECOND

DIFFERENTIAL

EQUATIONS. Linear One

Letter

General Exact

235

Equations,

"

236

237"238

Absent,

Equation. Removal Differential Equations,

of

Linear

.

.

Term,

a

.

.

239 .

.

.

.

"

240

241"242

CONTENTS.

xiii

CHAPTER

XVI.

EQUATION

DIFFERENTIAL

LINEAR

CONSTANT

WITH

COEFFICIENTS. PAGES

General

of

Form

The

Complementary

The

Particular

245

Function,

252"263 Constant

with

Form

Linear

to

264"265

Coefficients,

XVII.

CHAPTER ORTHOGONAL

Some

Important

266"269

Equations,

Dynamical

Illustrative

Further

EQUATIONS.

MISCELLANEOUS

TRAJECTORIES.

Trajectories,

Orthogonal

251

"

.....

Integral, Reducible

Equation

An

243"244

Solution,

270

271

"

....

272"277

Examples,

278"308

Answers,

ABBREVIATION. To

indicate

derived, in in

cases

the

common,

(a)

the

=

St.

where

which

from

sources

a

group

references

are

of

colleges

abbreviated

Peter's, Pembroke,

Corpus

of

many

examples

held

an

follows

:

have as

the

are

examination "

Christi, Queen's,

and

St.

Catharine's.

(j8) =

Clare, Caius, Trinity Hall, and

(7)

=

Jesus, Christ's, Magdalen,

(d)

(e)

=

=

Jesus,

Christ's, Emanuel,

Clare, Caius, and

King's.

King's.

Emanuel, and

Sidney

and

Sidney

Sussex.

Sussex.

CALCULUS

INTEGRAL

CHAPTER

NOTATION,

The

of

area

of

Aim

obtain

the

the

Calculus

Integral to

APPLICATIONS.

SUMMATION,

and

Use

1.

I.

the

is

plane

method

of

bounded

space

of

outcome

general

some

Calculus.

Integral

by

an

deavour en-

the

finding

curved

given

lines. In

the

area

it is

into

a

then

the

limit is

each

number

of

is

problems

such

the

volumes

bounded

moments

of

E.

i.

c.

that

the of

areas

up

obtaining

elements small

when

and

their

increased. when

discovered, as

of

these

an

elements.

small

very

infinitesimally

found

be

all

such

divided

space

method

some

of

sum

ultimately

summation

line,

the

of

form

to

infinitely will

It

have

this

suppose

number

large

of

determination

the

to

necessary

very

We

of

problem

inertia,

it

finding surfaces

by the

be

may

of of

them,

the

length

the

of

method

a

applied

given

positions A

such

once

of

shape

other

to a

curved

and

the of

determination of

Centroids,

etc. "E

CALCULUS.

INTEGRAL

2

Throughout the book all coordinate all angles will supposed rectangular, measured

in

circular

measure,

supposed Napierian, exceptwhen of 2. Determination Notation. be Summed.

an

Area.

and

axes

will be

be supposed all logarithms stated.

otherwise

of Series to

Form

of the portion to find the area Supposeit is required of space bounded AB, defined by by a given curve of and BM its Cartesian equation, the ordinates AL A

and

B, and the cc-axis.

L

0,0,0,0, Fig. 1.

Let

LM

QiQz,Q^Qv

be divided into n equal small "f lengthA, and let eacn """"

parts,LQV

and 6 be ?i/L Also if the ordinates a

Then b abscissae of A and ". a be the equationof the curve, (f)(x) y LA, QiPp $2^2*e^c-'through the several the =

"

=

points L, ^(a+K), ^(a+2A),etc. Qv Q2,etc.,are of lengths"j"(a), Let their extremities be respectively A, P1? P2, etc., and completethe rectangles AQV PjQg,P2Q3,etc. of these n rectangles falls short of Now the sum of the n small figures, the area sought by the sum etc. Let each of these be supposed 1, P1J22P2,

CALCULUS.

INTEGRAL

the

term

h(f"(a+nh) or

the

limit

is taken.

which

A0(6)

Hence

vanishes of

limit

the

when

this

series

also be written

may

f6 I

"t"(x)dx.

a

3.

This

summation as

be

sometimes

may

elementary means,

Definition.

the

Integration from

to illustrate

proceed

now

we

by

effected : "

Cb

Ex.

Here

1.

we

/ e*dx.

Calculate

have

evaluate

to

+ ea+"

+ ea+h Lth==Qh[ea

b

where

=

a

+

.

.

.

+ nh.

ea)-^-=e* =Lth^h^p\ea=Lth^(eb

This

-

-

1

"

"

e\

X

"

[By Diff. Calc. for Beginners,

Art.

15.]

r=n-l

/b

xdx

we

to find

have

2

Lt

("+rA)A,

where

r="

2(a + rh)h

Now

and

in the

=

limit

becomes

22'

2

/61 "$x a

diminished indefinitely

of

we

have

to

obtain

the

limit

when

h is

APPLICATIONS.

SUMMATION,

NOTATION,

5

"

b+ h

a

"'

a-h and

when

h diminishes

without

limit,each

of these becomes

II

b'

a

/"==*.* f*JL

Thus

J

.r2

b

a

a

Ex. 4.

Prove

ab initio that

/"

sin

We

now

ofo?

#

cos

=

a

6.

cos

-

to find the limit of

are

[sina

+

sin(a+ k)+ sin(a+ 2A)+

sinf a+n

"

\

sin

.

.

.

to

terms]A,

n

l- Jsin n2/ 2,

| *

This

expression =

cosfa

" -

J

cos

"

" a

+

(2n

-

1)-j"

-

2JJsin2

sm-

which

when

A is

small ultimately takes indefinitely cos

a

"

cos

b.

the form

CALCULUS.

INTEGRAL

EXAMPLES. Prove

by

that

summation

/ sinh sir xdx /

2.

cosh b

"

cosh

"

a.

3.

/b

OdO

cos

4.

Integrationof

As

a

further

sin 6

=

a.

xm.

example we

the limit of the

sin

"

next

propose

to consider

+ 1 not

being zero.

of the series

sum

h[am+ (a+ h}m+ (a+ 2h)m+. 6

i

h7

where

=

a

"

--

,

n

and

n

is made

[Lemma.

The

"

m indefinitely large,

Limit

of

fy v"/

I\m+l

I

yin +

_

%

2

-

"

1

is

m

+ 1

when

A

is

Aym

diminished,whatever indefinitely y finite magnitude. For the expression be written may

may

be, provided it

be

of

-1

y and

since h is to be

ultimatelyzero

we

may

consider

-

to

be

y

less

than

unity,and

we

Theorem

to

expand

therefore

may

/

^\7?l+ ( 1 -J--

J

apply

the

Binomial

l ,

whatever

be the value

of m+l.

(See Dif.

Gale,

APPLICATIONS.

SUMMATION,

NOTATION,

for Beginners,Art.

Thus

13.)

the

7

expression

"becomes

-x(a convergent series) y A is

+ I when

"m

diminished.] indefinitely

In the result

put

i/ success! vely

and

we

a+h, a+2h,etc....a + (n

a,

=

"

l)h,

get

l-am+l_

(

T

~

r, _

1 -

(a + n^

_

h(a+n-Ui)m for

adding numerators

or

for

a

numerator

new

a

and

nominat de-

denominator,

new

+ (a+ /t)w + (a + 2h)m+ fe[aw

.

.

.

+

(a+ n^l

or

+ (a+ A)m+ (a Lth=Qh[am

'

m+1 In may

accordance

with

the

notation

be written '6

b" 7

xmdx="

m+1

of Art.

2, this

INTEGRAL

8 The

letters

CALCULUS.

and b may providedxm does not

When

whatever, represent any finite quantities

a

is taken is necessary in the a

become

small exceedingly to proof suppose h an

6

=

1 and

xmdx=

a

"

in the

that

limit

if

"7

-

is

zero

for

V

0, ultimatelythe

=

and

x=a

and ultimately zero, it infinitesimal of higher

as

order,for it has been assumed all the values givento y. When

infinite between

+ 1 be

m

comes be-

theorem

positive,

o

or

This theorem

if m

oo

=

+ 1 be

be written

may

negative.

also

"r

as according

or,

which

is the

oo

the

"M4-'

former

by 1

and

limit

thing,

same

-Lstn=

differs from

negative.The

is positive or

m+1

is therefore also

"

-"?,

positiveor negative. The

i.e. by 0 in the

"

,

limit,

n or

oo

case

as m+1 according

when

m

+ l=0

is will

be discussed later. Ex.

bounded

1.

Find

by

the

area

the curve,

of the the

the parabola 7/2=4a# the ordinate x"c.

portionof

#-axis,and

NOTATION, Let

length c into n equal portionsof which Then if (r+l)th,and erect ordinates NP, MQ.

divide

us

is the

NM

PR when

the

is infinite of the

n

sum

Lt^PN.NM

i.e. where

nh

the

parallelto NM,

drawn

be

APPLICATIONS.

SUMMATION,

of such

required is the limit (Art. 2), rectanglesas PM area

or

c.

=

Now

[By Area

of the

=f

of the Ex.

Find

2.

the

4.]

=f

rectangleof which the are adjacentsides.

area

Art.

mass

of

a

extreme

rod

whose

ordinate

arid abscissa

densityvaries

the

as

of the distance from one end. with power Let a be the length of the rod, o" its sectional area supposed uniform. Divide the rod into n elementary portions each of

length of

The

-.

densitv

zero

is w-,

\

"

-

"

n

and

(r+l)th

its

element

from

density varies from

n

(7+la\m 1

of the

volume

)

Its .

mass

is therefore

coa**1-

( *

intermediate

)

and

**

between

end

the |

"

*"

to

INTEGRAL

10 Thus

the

mass

of the whole

CALCULUS.

rod lies between

and and

in the

limit,when

n

increases

becomes indefinitely,

ra+1

5. Determination

of

a

Volume

of Revolution.

requiredto find the volume formed by the revolution of a given curve about an axis AB in its own planewhich it does not cut. Taking the axis of revolution as the cc-axis,the figuremay be described exactlyas in Art. 2. The Let

it be

Fig. 3.

elementaryrectangles AQV P-fy^P2Qz"etc.,trace in and their revolution circular discs of equal thickness, of volumes "jrAL2 LQ19nrP^ Q", etc. The several annular portionsformed by the revolution of the portionsAR^^ P^R^P^ P2E3P3, etc.,may be con.

.

CALCULUS.

INTEGRAL

12

before

dividingas

Then

into

elementarycircular laminae, we

have

/cy^dx

re

/ xdx

4a:r

"

2

=J cylinderof [Or if expressedas Volume

I

4a?r

AN

radius

PA7' and

heightAN.

series

a

[c

=

[Art.4.]

dx

x

o

r = "

.

2a?rc2.]

[Art.4.]

2

Ex.

2.

revolution

Find

the volume

of the

prolatespheroidformed

of the

ellipse~+^2

=

by

the

the #-axis.

l about

*

Fig. 5.

Dividing as

before

coincide with

axes

the

into

elementary #-axis,the volume

circular laminae is twice

/ Try^dx. Now

-a2

-

x*)dx

a

which, accordingto Article 4, is equal to

5[a*.("-0)-^] or

and

the whole

volume

is

whose

APPLICATIONS.

SUMMATION,

NOTATION,

[or if desirable we may the sign of integration, as

obtain

the

without

result

same

13

using

EXAMPLES. 1. Find

the

the

ordinates

2. If the

volume

of the

formed

volume

the the

volume

x2+y2 the

of the

a2 about

=

of

areas

point

revolution the

2, the

the

the

by

the

as

the

the

the

of the

the

line

x

this

when

triangle =

a.

triangle

revolution

of

.r-axis.

figures bounded the

area

each

by

ordinate

of each

x

=

about

of

the

h ; also the the #-axis :

=

aty a

circular

distance the

disc from

of

point to

the

density at

centre.

prolate spheroid =

each

which

the

l ellipse^2/a2-f^/2/62

density at

the

by

7/3 a*a

of

mass

the

revolution

of

mass

varies

of

area

sphere formed

the

(8) 7. Find

,#-axis find

the

by the reel-shapedsolid formed that part of the parabolay^"^ax

#-axis, and

the

curves,

formed

8. Find

of Art.

y-axisof

the

(a)

each

round

1 revolve

of the

volume

latus-rectum.

6. Find

following

^^ the #-axis,and

"

#-axis.

the

about

5. Find

circle

method

the

the

revolution

the

y

solid formed.

about

by

curve

line y=x tan 0, the #-axis and of the cone formed also the volume

4. Find

off

the

the

Find

cut

Question

in

by

by

revolves

by

#=".

#=a, area

3. Find

bounded

area

about

be //x,

formed the

by the #-axis,supposing

II

CHAPTEE

METHOD.

GENEEAL

Before

6. the

theorem the

Calculus,

of

the

with

indicated

operation

of

general

a

enable

cases

many

applications

establish

shall

we

in

will

which

result

further

proceeding

Integral

FOEMS.

STANDAED

to

us

infer

by

n

I

"p(x)dx

a

without

recourse having difficult,process

often

the

to

of

tedious,

usually

Algebraic

and

Trigonometrical

or

Summation. 7. finite

b

difference

h,

so

limit

of

ike

h

when increased

[It be

the

may

that

b

a

"

of

sum

be

once

greatest

term

6,

"

into It

is

and

n

suppose

portions

required

find

to

a

the each

the

4"(a + 2h)+...

+

indefinitely,

0(6 and

-

h)

+

0(6)],

therefore

n

limit.

without at

be

a

values

series

diminished

is

let

nh.

=

the +

;

is

which

x

finite

given

divided

be

"p(a + h)

+

x

to

a

"

equal

ft[0O)

variable

the

of

function

any

between

continuous

and b of

and

"/)(x) be

Let

PROP.

seen

the

-

that sum

this

limit

is

is

a)"t"(a+ rh)

+

h"$"(a+

finite, for

if

"$"(a+rh)

x

b and

between

intermediate

Let

15

be \fs(x)

of

function

another

shall then

of

a.]

i.e.such its differential coefficient,

We

FORMS.

is finite four all values since by hypothesis""(#) finite,

is

which

STANDARD

METHOD.

GENERAL

x

such

is "j)(x)

that

that

that

prove

Lth^["fa)+^a+h)+^ By

definition

and

therefore

where

a:

a

=

limit

quantity whose indefinitely ; thus

is

diminishes

^a)*

a

is

zero

+halt

=\/s(a+ 7i) t/r(a)

h(j)(a) Similarly h"f"(a

h

when

"

etc., + nil) Ih) \[s(a =

"

where

the

"

quantitiesa2,

quantitieswhose

limits

a3,

...,

zero

are

\

an

all, like

are

av

h diminishes

when

indefinitely. By addition, + 0(a + h)+ "f"(a h["f"(a)

Let

a

be the

greatestof

the

quantitiesav

a2,

.

.

.

then

Afoi+ag+^.+On] and

therefore

vanishes

is

"nha,

i.e.

in the limit.

"("

Thus

"

a)a,

,

an,

CALCULUS.

INTEGRAL

16

limit zero; hence if desire,it may be added to the left-hand member this result,and it may then be stated that The

term

is in fc"/"(6)

the

we

of

1 "/)(x)dx\ls(b)\ls(a).

.e.

=

result

This

"

denoted is frequently \[s(b)\fs(a)

the

the form

of

p\/r(a3) J

notation

.

this result it appears function ^fs(x) (of which

that when

From the

by

"

is obtained, the coefficient)

is "/"(x)

of

process

differential

the

algebraicor

rb

summation trigonometric

I

obtain

to

avoided. The

"j)(x)dx may

be

a

supposed in finitequantities.We shall

the above work extend our now

I "f"(x)dx express

the limit when

letters b and

to denote

notation

to let

as

so

a

are

a

b becomes

i.e. infinitely largeof ^(6) ^js(a), "

I

(j)(x)dxLtb=x I "j)(x)dx. =

a

a

(j)(x)dx fb

understood

shall be

we

to

fb

-\I,(a)]

or

Ex. Hence

1.

if

The

and "df(x)":L^ ' m+l

coefficient of ^"

differential

we "$"(x)=xm

Lta=00\"f"(x)dx.

-

is

plainlyxm.

have

/

J

x

m+1

m

+ \

m+l

STANDARD

METHOD.

GENERAL

Ex. 2. The quantitywhose Hence known to be sin x.

FORMS.

17

differential coefficient is

cos

a?

is

"6

sin b

=

quantitywhose

The

Ex. 3. itself ex.

dx

x

cos

sin

"

a.

differential coefficient

is e* is

Hence

Ex. 4.

EXAMPLES. the values of

down

Write

rl

/V"dr, CiX) /b

1.

,-2

/V"cfo?, 3.

2. I X 2i,

X

Cf/JCm

0

a

I

o.

cfo, d/X^

X

4.

1

it

ir

/2 cos

rA

dx,

x

6.

/

r4

sec2^;dx^

7.

/

\

o

ia

8. Geometrical The

proof of thus be a

the

Illustration of Proof. above

theorem

may

be

interpretedgeometricall

:

"

of which the ordinate is finite Let AB portionof a curve and continuous all points between A and B, as also the at makes tangent of the angle which the tangent to the curve with the a?-axis. Let the abscissae of A and B be a and b respectively. Draw ordinates A N, BM. be divided Let the portionNM into n equal portionseach of lengthh. Erect ordinates at each of these pointsof division

cutting the

curve

tangents AP^

in

P, Q, R,

PQi, QRi, etc.,and

Draw lines

etc.

...,

the

the

successive

AP2,PQ2JQR2,..., parallelto

^r(x\and

y then =

the

and ,r-axis,

let the

equation of

the

V^') ""M" + Zh\ etc., are + h\ "$"(a "f"(a\ "$"(a respectively let

=

tanP.JPj, E.

I. C.

taii^Pft, B

etc.,

curve

be

CALCULUS.

INTEGRAL

18 and

-h),

...,

it is clear that the

Now

P2P, "2", R2R,

the lengths respectively

are

of

sum algebraic

is ...,

i.e.

MB-NA,

Hence

s,

u

L

M

x

Fig. 6.

portion within square brackets may be shewn for instance be For if R^ with h. diminish indefinitely PjP, Q^ etc.,the sum greatestof the several quantities the

Now

[P1P+Q1Q+...] if the abscissa of

But

Q

is

"nR1R,

i.e.

to

the

"(b-a)-}~.

be called #, then

(x)+

and

-^"(x+

Qh\

[Diff.Calc. for Beginners, Art. 185.] so

R^R

that

"(x4- "9A) =

(x+ Oh),

(6 a)

and which

=

-

is

an

infinitesimal in

generalof

the first order.

20

CALCULUS.

INTEGRAL

10. When we

are

lower

the

limit

is not specified and merely enquiringthe form of the (at present) a

function \fr(x), unknown differential coefficient whose is the known function $(#), the notation used is

the limits

beingomitted.

11. Nomenclature. The

of these

nomenclature

is as expressions

follows

:

b

r(p(x)dx is called the "definite" b ; a and

or

of "f"(x) between integral

fx I

"j)(x)dxor

\{s(x)\[s(a) "

where a

the upper limit is left undetermined "corrected" integral;

"f"(x)dxor without as

the

calculus

limits

is called

-^(x)

limits and regarded merely specified any of reversal of an operation the differential is called

"indefinite"

an

"

or

unconnected

"

integral. 12. Addition

of

a

Constant.

the differential coefficient of ^]s(x\ it is also the differential coefficient whatever of \lr(x) C is any constant + C where ; for is zero. the differential coefficient of any constant

It will be

Accordinglywe

This

constant

obvious

might

if

that

"p(x)is

write

is however

not

usuallywritten clown,

but

GENERAL

METHOD.

will be

understood

STANDARD

in

exist

to

FORMS.

21

definit all "cases of in-

integration though not expressed. of indefinite processes frequentlygive results of different

13. Different will

Idx }*/I-x2

instance is the

,

is sin'1^

what

differ by

is a

cos~%, for

"

;

reallytrue

for

,

Vl pressions. ex-

cos'1^.

"

is that

sin"1^

and

"

cos"1^

constant, for

f

that

so

form

differential coefficient of either of these Yet it is not to be inferred that sin"1^^

But

or

integration

1

dx

"7

sin ~lx

=

Vl-a2

or

,

dx=

"

cos-^

J/s/l-a;2 the

constants beingdifferent. arbitrary

14. Inverse

Notation.

verse Agreeably with the acceptednotation for the intions, and inverse HyperbolicfuncTrigonometrical we might express the equation

j5^)

or

and

it is

useful occasionally

=

^); to

employ

this

notation,

INTEGRAL

22

CALCULUS.

character which very well expresses the interrogative of the operation are we conducting.

satisfied

Laws

15. General

by

the

Integrating

Symbol \dx. (1) It will symbols that

plain from

be

the

meaning

of

the

s

but that

constant. is "j"(x) + any arbitrary l-y-(fi(x)dx

is distributive; operationof integration

(2) The for if u, v,

functions of

be any

w

x,

-T-j |u^+l^^+l^^r and

therefore

constants) (omitting

\wdx l JurZ#+|i;cfe-f =

is commutative (3) The operationof integration with regardto constants. (I'll

For

if

-j"

=

v,

and

be any

a

d

constant,we

have

du ,

so

that

(omitting any

constant

au

or

which

=

of

integration)

\avdx9

a\vdx=\avdx, establishes the theorem.

METHOD.

GENERAL

several

Integration of

-

d

"

xn+l nrfll

-

dx been

(as has Art. 7, Ex. 1)

Thus

the of

the index

For

+ l

n

already

seen

in

Art.

4

and

increased.

so

example,

/nA X

X

r =

5

Q

r

11

~"''}X x=if

;

}x

EXAMPLES. TTn'^e down

the

of integrals 1.

O

x

^"J

1

0

""}

b

a

O.

in

of any constant integration the index by unity and divide

for the

rule

is,Increase

x

obtain

we

^

+ 1

_

_

Hence

of

xn.

n

by

23

now

By differentiation of

power

FORMS.

proceed to a detailed consideration forms of functions. elementaryspecial

16. We

17.

STANDARD

"#"

^7999 #1000.

-^"Jf

c

x~^

T4=

1

"4^4-

INTEGRAL

24

18. The

Case

CALCULUS.

of x~\

It will be remembered

that x~l

or

is the

-

differ-

x

ential coefficientof

Thus

logx. fl

\-dx

logx.

=

Jx This

therefore

forms

apparent exceptionto the

an

generalrule f

^n+l

\xndx =

19. The

result, however, may be arbitraryconstant, we

deduced

Supplyingthe

/xndx

C

+

=

=

A

~

is still an

Taking the

-I-A

"

"".

+ l

n

and

limitingcase.

n+l

C+

=

a

have

n+l

where

as

constant. arbitrary

limit when

-

l=0,

+

n

takes

-

the form

logx,

[Diff.Calc. for Beginners, Art. 15.] and

as

C is

we arbitrary

infinite

portion

may

suppose

togetherwith

-

7i

it contains

that

-

another

a

tively nega-

arbitrary

~\~J.

portionA. Ltn==-i {xndx logx + A.

Thus

=

20. In we

the

same

in the

as

way

have 1

(n + V)a(ax+ b)n

=

and

of integration

+ 6) ^-log(a% "v

= "

xn

and

STANDARD

METHOD.

GENERAL

f/

therefore

IA

'

=

"

"/

y

I '"r Jax + b

printed as

a

I -r,

(n+l)a

shall

we

"

Jax

,

-\og(ax + b\ 6V

=

fFor convenience

often

dx

,

Jja2+x*

+ b'

25

(oo5+ ")n+1

7

\(ax+o)ndx

and

FORMS.

as

f

1

Jax

+ b

find

I J*Ja*+ ,-

"jdx

"

o,

etc.]

x*

EXAMPLES. Write

down

the

1. ax,

integralsof

of1, a+x,

x

"

a"x?

x,

1

x_ a+x a? x

a

2

a

'

a-\-x

3. a

+ x

a+x

21. We

may

coefficients of

a

"

bx

a-x*

next

(a-#)2" (a x)n* "

x+a

x

remark

and ["f"(x)]n+l

d

"

(a+x)2

(a

"

xY

that since the differential of

log$(x)are respectively

and

{["t"(x)]n"t"'(x)dx

have

we

=

and

The

is of great especially the integralof It may thus : be put into words use. is the differential any fractionof which the numerator is coefficient of the denominator log (denominator). second

of these

results

"

For

example,

INTEGRAL

/ /co\,xdx J

CALCULUS.

xdx

*"

=

Sill

/tan .#

dx

=

I

"

"

log sin

#,

log cos

x

X

"

J

=

.

-a?^

=

"

log sec

=

x-

cosx

EXAMPLES. Write

down

the

integralsof

(a

nex,~-, ~\~Ct

"

22. It will the that

perceivedthat

the

of operations

IntegralCalculus are of a tentative nature, and in integration ledge success depends upon a knowof the results of differentiating the simple

functions. of

be

now

standard

It is therefore forms

the practically

which list

same

and differentiation,

the

necessary is now

to learn the table

appended. It is already learnt for

that proofsof these as

results lie in

of the the righthand members differentiating results. The list will be graduallyextended supplementarylist givenlater. PRELIMINARY

TABLE

OF TO

RESULTS MEMORY.

TO

BE

several and

COMMITTED

a

INTEGRAL

28

them. as

x

The

For

it is

the

through therefore

are

Also

is obvious.

reason

increases

and

instance,x

Each

first

of these

functions

; their

quadrant

negative. help to observe

further

a

CALCULUS.

the dimensions

being supposed linear, /

a

"

i

dimensions.

There

2 d~"

tan"1-

-\-X

be

integrationmust be

be

no

"

"

a

"

is of

side. zero

X"1

prefixed

-

dx

C J

therefore

v

"

of each

efficients co-

to

the

in-

a

/

tegral. Again

not

could

"

decreases

differential

2

is of dimensions

of dimensions

(which

is of

Hence

-1.

-1.

Thus

dimensions).

zero

the

the result of

integralcould student

The

should

a

therefore

have

factor

is to be

-

in remembering difficulty

no

in which

cases

the

prefixed.

a

EXAMPLES. Write

down

the

indefinite

integrals of

the

tions following func-

:

"

'

2.

3.

*

cos2-,

coss# .

sin #,

2 4.

cot

x

+ tan

x,

cos^f

-+-^-

.

\sin^7

#e + e*

'

log sin

x

V

snr^/

^sec-1^.

\/^-i*

CHAPTER

III.

OF

SUBSTITUTION.

METHOD

The to

V

by

z

being if

Or

the

any we

formula

the

To

variable

independent

prove

change

x

of

formula

x.

be

this, it is only

necessary

to

\Vdx\

=

=F.

then

du

du =

--

dz

whence

the

changed

V=f(x),

write

u

But

be

may

F(z), by

=

function

will

Variable.

Independent

the

of

Change

25.

u

dx

dx

dz

I =

J

~rrdx

V-j-9

"-_-=

-j-.

dz

dz

write

from

x

INTEGRAL

30

Thus

to

CALCULUS.

integrate/

-dx, let tan~1#=;s.

Then

dx

1

n

and the

becomes integral

*" dz

26. In

after

formula

usingthe

choosingthe

it is usual

to

form

make

of the transformation x F(z\ of differentials, writingthe =

use

equation

j^=F'(z)dx as

the formula

Thus

be

will then

of the left hand in the

side

dx reproducedby replacing and x by F(z). by F'(z)dz,

we precedingexample,after puttingtan~1^ti=0,

write

may

*=d* -

and

I+x*

27. We is

F'(z)dz]

=

a

next

definite

The

one

l+.r

consider the between

result obtained

case

when

the

above, when

x

=

F(z) is

Let then and

if the limits for

a;

be

integration

limits. specified

a

and

b,we

have

OF

METHOD

Now and

when

when

SUBSTITUTION.

x

=

a,

z

=

F~

x

=

b,

z

=

F~

31

\a) ; \b}.

f{F(z)}=-j^,{F(z)}

Also

and whence

with rethat the result of integrating gard f{F(z)}F'(z) limits F~\a) and F~\b) is identical to z between with that of integrating f(x)with regardto x between

so

the limits Ex.1.

a

and

Evaluate

6.

/

x"z^^ and

Let

^x Ex.

2.

Let

^3=2;,and

Evaluate

/.

Put

^?=tan

\Txdx.

N/a?

therefore

cosfjxdx"

-

cos

-

J

dx=Zzdz

I -cos2. J z

/.Aos

=

/cos

2-2(^2=2

z

dz

=

2

smz

J

x^dx. 3xPdx=dz

therefore

/^2cos x*dx

;

;

llcoszdz ^smz=^ =

e, then dx=sec20d6

;

when

#

=

0,

we

have

0

when

#

=

1,

we

have

0=.T ;

=

0, 4

sin x\

CALCULUS.

INTEGRAL

32

ir

f-T^=-dx \J\+*

:.

P *?"| sec20 dB

=

fsec0 "

"4 Let

6^

=

^, then

dx

f

Evaluate

sec

0

sec

-

=

V2

1.

-

jo

dz.

=

=

0 dB

{

\ Tsech^^]. [i.e.

_x

x

e'

exdx

B tan fsec

=

sec#

{ =

Ex.4.

ir

When

when

#=0, 0=1, and

Hence

3=e.

rtan-'/V= ten-

=

2

""

tan-1

-

=

t

Ji

L

"

o

The

indefinite

is tan~V. integral

EXAMPLES. 1.

Integrate

(Put

excosex

-

cos(log x)

^=4

(Pat logx

=

4

x

2. Evaluate

\-=-,dx (Put x*=z\ "

J 1+^4

reintegrate+ fl ^ Evaluate acos#

-

-,

a^sin

I*

"

J l-f#6

(Put a*=z). v

ex + b tanh

x.

(Put ^+1=4

"

a

Q

5. Evaluate

/"

6. Evaluate

/

7. Evaluate

dx -

a*

/*"

(Put x-\=z).

"da?

J 2V^(1+^) 8. Evaluate

9. Evaluate

[

-1

dx.

/ J 2W#

dx. -

1

(Put

(Put ^=02).

#

METHOD

NOTE

HYPERBOLIC

THE

ON

SUBSTITUTION.

OF

33

FUNCTIONS.

28. Definitions. it is desirable that the purposes of integration the definitions and student shall be familiar with fundamental propertiesof the direct and inverse hyperbolicfunctions. By analogy with the exponentialvalues of the functions sine,cosine,tangent,etc.,the exponential

For

"

e-*

"

_

PTP

__

2

2 are

ex-e~x

ex+e~x _

~

e?+e-*'

written respectively cosh x,

tanh#,

29.

Elementary Properties.

We

have clearly

coth

~

C/

-,

tanh

x

=

x

=

V

X "C"JLJ.iJ.l

e~x

sinho?

etc.

-

"

t"rihx

-=cosh

^

2 sinh

x

cosh

x

=

2

"

:

^

"

"

.

-^"

=

^

with many other results analogous to the formulae of Trigonometry. E.

i. c.

=

sinh2#,

"

AM

c

common

CALCULUS.

INTEGRAL

34 30. Inverse Let

us

function

Forms. for

search

the

of

meaning

the

inverse

sinh"1^.

Put t

then

x

=

smh

y

and

"

Thus and

=

y we

shall take this

=

x"

=

log(x"

expressionwith

a

sign, positive

viz.,log (" + "v/l + #2) as sinh"1^. 31.

Similarly, puttingcosh~1x

y,

=

have

we

ty+e-y x

and

"

cosh y

==

e*y

and

ey

=

whence

y

=

and

"

JL

we

shall take this

x"*Jx*-l, log(x " *Jxl 1), "

expressionwith

a

positive sign,

viz.,

/#2

1

"

32.

Again,puttingi"nh-lx x

and

as

=

tanh

y

=

y,

"

have

-

e2y=-"

therefore

1"

whence

we

tanh

-

lx

=

x

4-log S ^ "

1"

"-

05

INTEGRAL

36

cu-l

x

whence

CALCULUS.

tan-

=

0, 2

and

tan

a;

eu-l

x

tan2 2"

=

=

^TTT

0_e-

.e-

!~4^-: ~2~ Hence

^

=

tan

~

1sinh u

=

gd u.

logtan(j+|)=gd-^

Thus

of

the inverse Gudermannian

x.

EXAMPLES. Establish the 1.

2.

results : following

"

/cosh#cfo?=sinh#.

4.

/sinh

5.

xdx

cosh

=

x.

/sm,.^o?.r

=

"

sech

x.

J cosh%

J 3.

/cosech2.rc^= -coth#.

6. (sech2A'dr=taiihtf. J

7. Writing sg results :

x

for sin

gd x, etc.,establish

"

(a) /

the

following

SUBSTITUTION.

OF

METHOD

37 ,*

36.

and

Integralsof

-4-f

X

differential coefficient of

The

loge"

-

*

.

log"

=

z

..

Jx/^2-^2 the

resemble

log

=

a?

"

--

the

38. We

,

established

put

=.

t

cosh

a

=

Hence

u

=

W^2

+ tt2

du

oj

leZu t6 =

J

"

u,

cosh

a

f =

a

u

integrate *A2-tf2. dx

=

u.

=

a

sin 0 ;

=

a

cos

have

we

du

-

"

a?

az

sinh"1-.

=

Integralsof

then

:

a

=

Let

a

the results thus

\/x2+

and

=

39. To

viz.,sin'1-*

sinh it, then

a

=

Similarly puttinga? a cosh Fa sinh u f dx I!" ri Jx/x2" a2 J " sinh /

^,

results

dx

)*Jx*+a^ dx

these

aid to the memory.

might have

f To find I

,

a

integral I

for the

analogyis an

.

a

Va2-*2 and

1 l-

^ =cosh-1-.

hyperbolicforms

inverse

that

-

a

\/x2

+

x

,

,

37. In

,

a

dx

F .,

0.

bimilarly

smh

=

"

0 d$,

is

7 laK

J

11^

=u

=

cosh~1-. a

INTEGRAL

38

CALCULUS.

{+/tf-^dx

and

=

ia sin 6

.

0+

cos

a

or

-

-^

sina

40. To

integrate

Let

cc

cf^

then

sinh z,

=

a

=

acosh0

=

cosh2z,

=

a2\ cosh2z

,

then

we

1 + sinh20

since

have

IJ^^dx

=

|a sinh

0

dz

.

a

cosh "z+ -^-

.....

.

Va2

.

,

2-smh-1x

OF

METHOD

41. To

39

integrate

Let

x

dx

then then

SUBSTITUTION.

since

cosh%

1

"

JJsif^cPdx

a

cosh z,

=

a

sinh

=

sinh20,

"

dz ;

z

a2 sinh20 dz

j

=

C62?

sinh

Ja

=

\2

z

a

.

"

^-,

-log-

or

we

0

a2!

a

"

cosh

42. If have

we

put

tan#

therefore

", and

=

__

[by Art. 40.] tan

sec

x

05

-

,

h

"

^

snce or

_

-

_

2 cos2#

+

,

J log^ n

.,

J log(tana? +

-,

"1"

sec

^),

INTEGRAL

40

43.

Integralsof

Let tan

^ 2t

z

=

;

CALCULUS.

cosec

and

x

sec

x.

differential takingthe logarithmic

1

9x

dz =

or

"

^

2

x

dz

dx

7

^dx

-

z

-;

"

"

.

z

smx

n,

2tan2 xdx"\

Icosec

Thus

In this

example let x dx

Then and

sec

Isec xdx

Hence

44. We

have

=

2*

logtan

(T+ 9)-

=

logtan

(r+s)

or

STANDARD

a

dx

,x

=1"g

JTP^l

l+Jx2 a2dx "

*x-

FORMS,

g

Jx/^+o1

I+/x24 a2 die

~

x+\/x*+a?

dx

"

""

the

now

\\/a2 x2dx

^.

+ y.

logtan

. ,

f

=

=

ADDITIONAL f

-=

logz

dy,

=

ydy

=

"

=

=

fi2 =

2G"

a

OF

METHOD

Icosec

dx

x

=

SUBSTITUTION.

logtan^.

Isecsccfo =log tanf^-

EXAMPLES.

3.

the

down

Write

of integrals

x

x

4.

5.

7.

1 8.

cosec

2#, cosec(a#+"),

^'

cos

tanV

3sin^-

x

-

"/

Find

sin ^7+ 6

1"

/cosec#cfo?=logtanby expressingcosec

10. Deduce

11.

a

-sin2^'

^j

\SQexdx by puttingsin

x=z.

x

as

CALCULUS.

INTEGRAL

42

that

Show

12.

/ /

cosh

dx sec

x

=

Integrate

13.

1

tflogtf'

when

lrx

represents

log

log

the

log

log

^7,

being

repeated

...

times. r

15.

Prove

[ST.

PETER'S

COLL.,

etc.,

1882.]

CALCULUS.

INTEGRAL

44

with

\dx integral 0'(#) lx/r(#)cfo

new

a

which

be

may

easilyintegrablethan

more

the

original

product. be put into words thus rule may Integralof the product "j"(x)\{s(x) 46.

The

Integralof 2nd Integralof [Diff.Co. of

1st function

=

-the Ex.

1.

another

cos

important

integralin

by

1st x Int. of

2nd].

possiblejxcosnxdx

with

nx.

to

which

if

connect

the factor

has been

x

if x be chosen the be done as may second integral"$(x\ i.e.unity,occurs

Thus

"

x

Integratex it is

Here

:

removed.

$(x\ placeof x.

function in

This

since in the Then

the rule

/"l.5

(xvxnxd**,*!*"?J

'

J

n

sin "

x

If

9^7

-

cosn"N

n\

n

sin

n

cos

nx

I

-

"~~~~\

/

nx

'

Unity may integration.

47. an

Thus

be taken

as

/1 logx

/logxdx"

of the factors to aid

one

dx

.

=

=x

x

logx log x

"

"

/x

"

I \dx

-(logx)dx

TION

INTEGRA

repeatedseveral f 9 / #2cos

Thus

PARTS.

45

operationof integrating by parts may

48. The

mt.

B T

be

times. #2sin

? dx

nx

f sin / 2#

nx

0

"

"

-

nx

7 dx.

-

J

J

n

n

and /

n

finally,

f J

$x^ J

Hence

=**** "

dx

nx

-*\_^COS nL

n

#2sin

Zx

nx

n

cos

I

_

~

2 sin

nx

nx

7^

T9

*

of the subsidiary into 49. If one returns integrals form this fact may be utilized to infer the the original result of the Ex.

1.

integration.

/eaxsin bx /eaxcos

and

if P therefore,

and

/eax$m we

have

and whence

aP

dx

=

"sin

bx-~\

e^cos

bx dx

=

"cos

bx+-l

e^sin bx dx ;

Q

stand

bx dx

+bQ

dx,

for respectively and

=

bx

/eaxcos

eaxsin

bx

dx,

bx,

-bP+aQ=eftxco8bx" nrct

n P=eax-

sin bx =

"

b

cos

bx

"

r?

a2+o2

/

and

w+v "

(a2 +

(bx 62)~Yeaxcos \

"

tan"1

-

aJ

).

INTEGRAL

46 The

should

we

will observe

student

that

by puttingn=

obtain

CALCUL

US.

these

results

"

the

are

same

that

I in the formulae

^)""ss""it^"^(^+"^)' [Diff.Gale, for Beginners, Art. 61, Ex. 4.] And

this

obvious.

otherwise

is

rxsm/j^,\ jg fag

game

to

as

the

increase

multiply by r

/

pnsi^

""

angle by tan"1-, the

if to

For

factor

a

differentiate

Va2 +

62 and

to

""

is the

which integration,

a

inverse and

diminish

Ex.

divide

operation,must the

angle by

Integrate\/a2

2.

"

xl

out

again

the

factor

Va2+62

tan"1-.

by

the rule of

by parts. integration

r _

,

_

J A/o2^2^=

[Note this step.] c

%

a2sin~

l-i

CL

Iv a2

-

"

J

and dividingby 2, whence, transposing

which

Ex. Here

agrees 3

with

the result of Art. 39.

Integratee*xsm2x e3xsin% cos3#

cos3^.

"

=

"

-=

-x _(2e3a:cos -

"(1

"

cos

4#)cosx

TION

INTEGRA

Hence, by Ex.

"\

"

-j=

cos

1r j

tan~

16, p. 55, Diff.Gale, for Beginners,putting

Ex.

l in the

"

^34

3\/2 ,: V

"

fx

cos(3^-^--^cosf5^-tan-1|) V 3/J 4/

J_

n=

47

^

-

[Compare

TS.

PAR

1, cos3# dx

I e^siiA

B Y

result.]

EXAMPLES.

Integrateby parts : 1. xex, x^e*,xze?)x cosh #, ^?COS2 2. ^?COS^7, ^2COS07, sin

sin

sin 2# sin 3^.

3.

x

4.

#2logtf, ^nlog^7,^n(log^)2. e^sin^costf,e*sin x cos ^ cos 2#. eaxsin^ sin qx sin r^?.

5. 6.

x

cos

^

#,

7. Calculate

x

|^sin^^,

0

0

8. Show

9.

Integrate Isin"1^^,/^sin"1^^, \

Geometrical

Let

PQ

and

Illustration.

referred to be any arc of a curve Ox, Oy, and let the coordinates of P of Q (xv y^).

Let

PN,

be

QM

area

PNMQ

the

ordinates

and

rectangular be (XQ,yQ),

PNV QM1 pointsP, Q. Then plainly

abscissae of the

But

0

that

50.

axes

/*x sin2^pc?^1,/

=

rect.

area

OQ

-

PNMQ

rect. OP

=

f

-

area

the

48

INTEGRAL

and

area

CALCULUS.

PN^M^Q

cv\

ri

Thus

Let

us

o

o

consider the

now

I x dy.

=

curve

J

to be defined

by

the

equations and and

y let

the values of t corresponding to #0, y0, and a^, 2/1 of cc and y respectively.

t0and t"be

the values We then have

ri

ri

I *0

and

and

7

,

r*i

2/"x^=l vdu=\ *"

"0

I o?c?2/=l udv**\

TION

INTEGRA

that the

so

equationabove

and thus the rule of

B Y

may

PARTS.

49 '*

be written

by partsis established integration

geometrically. 51.

Integralsof

Ia^sin Reduction

the

Form

dx, I#mcos

nx

nx

dx.

formulae

for such integrals the above as readilybe found. Denote them respectively by may Sm and Cm. Then, integrating by parts,we have at once cos

m~

nx

and(7m= Thus

and

Om= ,sin7i#

cosnx

____+m^

=

and

771(971 v___ "

, "_

Cm=

"

---

m(m

,cosnx

-l"-

----

1)

"

"

n

Thus and m

the four

when =

l

the

found, viz.,

are

n

"Sn "

E. I. C.

for integrals

^

I sin

cos

me

nxdx=

J

,

n D

cases

m

=

CALCULUS.

INTEGRAL

50

sinrac

f

xv

GT0 =

at. o\ =

I

7

cos

nx

dx

nx

dx

sinnx \"

eosnx

Icesin

t

\ cos \x J

=

,

7

,

=

x

"

"

^

~

0

=

T^CC ax

=

by

be deduced

all others

can

the above

formulae.

Integration by

for

Rule

the

of

52. Extension

of applications

successive

Parts. If

and

u

be

v

x

we

rule for

denote

with respect integrations of the extension prove the following may by parts, integration

\uvdx

where

dashes

and

x

suffixes

differentiations and to

of

functions

=

uvl

"

u'v

for

u^n~1^ is written

\uvdx

u

with

=uvl

TI

"

1

dashes; for

\u\dx,

"

Vufv^dx =u'v2 "\urfv^dx, \vtf'v2dx =u"vz

Iu'"vBdx ufv^ =

etc.

Iu(n l)Vn_1dx

=

-

=

Vuf'Vzdx,

"

"

Iu^'v^dx,

etc.

u(n~ Vvn

-

Ivf^Vndx.

CALCULUS.

INTEGRAL

52

have

we

bx dx

Ixneaxsm

=

eaxsin (bx

"

J

d")

"

eaxsin(bx

"

r2^

r

"

^^~

r3

n\

eax{P sin bx

or

Q

"

cos

where X COS

-

3-

Q=

sin 0

"

n

"

...

xn~^

xn~l

xn

30"

"

^-

sin

20

+

n(n

"

1)" ^- sin 30

"

...

Similarly

L^a*cos ix Ex.

1.

Since

we

have

dx

=

eP*{Pcos bx+Q

sin

bx}.

Integrate ix^smxdx. \e*smxdx

S^e^sinf.r

"

-^J,

f^3ea:sm^^=^32'^ea;sm('.r ^ 3^22~Vsin^ -

2" .

-

-

VVsinf ?" 4 .77

-

\

6 .

2~VsinCr- TT)

/

=etc.

Ex.

2.

Prove ^|

-rjQ ^s

^Vto^ito-^-iy^j^^ /r=n EXAMPLES. 1.

Integrate (a) femai"~lxdx.

(5) (sfaitr^xdx. (c)

(d)

/"

(e) \

Ixv"Pxdx. '(/) /"cos-1^.

TION

INTEGRA

2.

Integrate (a) [x

(")

PARTS.

B Y

dx. sm"1f

(c) /sin-1'

/^5^"'. tan

(d)

lx

-

r

/ptn

-dx.

~

53

pin tan

-

lx

dx.

(c) J

*-*

dx.

4.

../*..

Integrate (a)

I

(b) I xefsm^x (c) I cosh 5.

/log

Integrate

-

J

6.

Integrate

7.

Integrate

8.

Integrate

9.

Integrate(d)

.

...

e(suix + cosx)ax.

\x

(e) I^22*sin

dx.

(/) /cos

sin bx dx,

ax

r ..

(a)

b log"\dx. -j

sin'1^ dx.

x

/cos201og(l+tan 0)dO.

J*4"|^.

^^

TKJPOS) 1892"]

(")

[a, 1892.]

1-cos^

i

/\

T-"

10. Prove

11.

i

that j

Integrate

2." dx.

/" c?2v

/u

7

du

c?v

2dx=u

"

"

"

/(asin%

+ 26 sin

v

x

C d^u

cos

i

/ v-"dx.

+

"

x

+

c

cos2^)e*^ [a,1883.]

INTEGRAL

54

12.

where

Show

the

that

if

u

be

series within

CALCULUS.

rational

a

the

integralfunction

brackets

is

of x,

finite. necessarily

[TRIN. COLL., 1881.] 13.

and

If

Ieaxcos bxdx,

u"

that

14.

Prove

v

Ieaxsm

"

bx

+ v2) (a2+ "2)0*2 =

dx,

prove

that

era*.

that

-"

m+1 Also

m+L

that

(m+1)2

(-ir-^!?

3

^"-1 where 15.

I stands Prove

for

logx.

that

(i.)

{e^w J

+^""j)"2 leax^n-'2bxdx. J a?+ri2b'2

[BERTEAND.] 16. Evaluate

/x* log(l x^dx, -

iT5

+

277

+

3T9

+

and

deduce

that

-==9""310ge2'[a,1889.]

V.

CHAPTER

FRACTIONAL

ALGEBRAIC

RATIONAL

FRACTIONS.

PARTIAL

FRACTIONAL

ALGEBRAIC

54.

\

-

-"

of

Either Fractions.

FORMS.

of

Integration

or

^

and

-"

forms

these

-9(x"a\

-

*"2

a?

"

FORMS,

should

be

thrown

into

Thus

=___

a2

x2

_

2aj\x

"

1, =

a^

log

s-

^

Ja2"

#

a

F

1 =

-

;"

4-

+

x

a

"

"

"

2a

f

a

"

L

a

i^"l .1 coth"1

"

a

J

a

=!f(-J-+_J_Yfo 2aJ\a4-a)

a;2

1

a

"

F

a+oj ,

=

^"

2a

x/

l,i

T^l

=-tanh"1-

losr"

.

toa

"

a;

La

aj

Partial

CALCULUS,

INTEGRAL

56

(Compare the

brackets

of the results in square

forms

1 .

for

the result before tabulated

with

C

dx

!~n

1 =

"

Jcr+or

viz.,

-=

x\

)

tan'1"

-

o

a/

a

dx

Integrationof

55.

f-H

Let

6

J

a

dx

=1f.

f

J

a

c

2

AV_^2a/ 2J "

a^a

\

4a2

dx or

2

take

we

as

Thus

the former b2 is " or

if 62

"

"

=

"

/

--

;

.

tan

"

"

l"

cot

-.-

~

differ at most by constants,but expressions givencase a real form should be chosen.

These

cording ac-

4"ae, I

or

arrangement

4ac,

7_

If b2

latter

4ac.

"

coth"1"

or

any

the

or

in

56.

FORMS.

FRACTIONAL

ALGEBRAIC

RATIONAL

of of expressions Integrals

the form

57

*

px + q

be obtained

can

at

px + q

tion followingtransformapb 2a (2ax+b)

by

once

_p

the

,

~~

the

of integral

the first part being

^" log(ax2+bx+ Za and

that

c),

part beingobtained

of the second

by

the last

article.

[The beginnershould obtained.

the above

notice how

form

is

of the coefficient of the differential

It is essential that the numerator

Jirstfractionshall denominator, and

be the that

all the

#'s of the

numerator

therebyexhausted.]

are

T? '

=

57.

+ J log(^2

4*7 +

5)

-

Although the expressionpx

2

tan-1^ + 2).

+ q may

be thrown

into the form

we by inspection, might proceedthus

Let where

:"

pa?+gsX(2oaj+6)+/i, X and

/x are

constants

to be determined.

by comparingcoefficients,

giving

X

=

and

pb =

--

Then

INTEGRAL

58

CALCULUS.

EXAMPLES.

Integrate 1.

f

4.

f fo+1)^

5.

/Jfl-LZ-^p.

xdx .

2

/"

^a^/7/y.

//y"

.

/ 0^t1

c^

"

6.

58. General Fraction and Denominator. of Expressions

\2

I

J x2+?

x* + 2x+l

3.

f/v"

Rational

with

the form

A~4,

Numerator

f(x) and

where

"/"(#)

9w

rational

functions of x, can be integralalgebraic by resolution into Partial Fractions. integrated of putting such an The method expressioninto Partial Fractions has been discussed in the Differential Calculus forBeginners, Art. 66. When the numerator is of lower degree than the denominator the result are

consists of the A

And

A

sum

of several such

Ax+B

terms

as

Ax+B

and

the numerator is of as high or higher degreethan the denominator we may divide out until the numerator of the remainingfraction is of lower when

degree. The terms of the quotientcan in that be integrated and the remainingfraction at once be put into Partial Fractions as indicated above.

case

may

A

Now at

once

fraction of the form any partial into A log(x a).

"

-

integrates

-

A

Any

fraction

of the

form

-.

ix

1 r"l

"

"

^~*

^ a)

A

(x"a)r~v

into integrates

CALCULUS.

INTEGRAL

60

integralis

and the

Ex. 3.

^

/

Integrate

Put

x

\

"

the fraction becomes

Hence

Dividingout

until

=

y3 is a l+

2y

1311111

and

therefore ^2

and

the

"We

now

a?

1

is integral

Ex.4. Let

1

Integrate =

1

+y

divide

?/.

Aj/

factor of the

the fraction

Hence

=

-dx.

; then

out

by

remainder,

ALGEBRAIC

RATIONAL

is a factor of the remainder. until ty4 coefficients : detached use 2 + 3 + 3 + 1

To

shorten

the

work

(J

) 1+2+1 l+

FORMS.

FRACTIONAL

f+f+

|

i-i-i j+t+ f

i

+

-f-f-i f+"+f f +tf +if

+

A

tt-A-A 551

ll-5y-

e

Now and

11

by

Rule

-5j/-5y2

=

ll-

5(^-1) -5(^-l)2

Gale, for Beginners, 2, p. 61, of the Diff.

5#2

\(x)

1

and 3

3(^

+ 1

3

l+#

3

x

Thus I!

^.2

-L

i

ijj

^

~

+ 1) 2(^7 1)4 4(tf I)3 8(^7 1)2 (a? 1)4(^3 5

^-1) and

the

-

-

-

-

integralis plainly

1

48

A7

1

1

+ 1

6

(2a?-l)-3 ^2-

61

we

INTEGRAL

62

CALCULUS.

EXAMPLES. 1.

Integratewith

v11*'

w'

2

^+ (iii.)

regard to

x

the

followingexpressions:

\V11V

T~\*

a)-1^ + b)~\

~f~

(viii.)

\7

?T7

"a*--") "

(iv) -/

"

"^

2. Evaluate

3.

Integrate (i) W

4.

dx f J (^2+a^2+62y

(iii) "'

f

J

Integrate

(xd* (i.) v ; J^+^2

.

+ l

do?.

[' (iii.) v

-

J^+l

(iv.)f

cto. Aa?2"t1

J^4-^2+l

(v.) r (vi.) /"(^-

\'

5.

FORMS.

FRACTIONAL

ALGEBRAIC

RATIONAL

Integrate /.

xdx

v

.

.

dx

x

dx

(vii.)

(^"T4)-

^

(iiL"

^"**"

W

("") (x\ VA*/

6.

~(~"

lX-^-4)' (x*+ i\/ i

Integrate J~.

~3,J~*

(VI.) -7 x""

d^t?

\

f

~t

\

"/

/T

j

j ^

\o/i

~t

""

:

\

V*-'"^*^

o\'

(viii.)

''

J

+ iy (#-l)2(#2

/Vtan~"^(9 and P\/c

7. Evaluate

8. Obtain

the value

c

o

cos

9.

x

dx

Investigate

10. Show

*

^2

that

r. fa "o

_f^

+

1)3

63

CALCULUS.

INTEGRAL

64

that

Prove

11.

dx

[+*

_2?r 2?r

a

b

+

~~

J

(x*

"ax+

"

"2X^2

bx

+

b'2)

V3

ab(d"

[COLLEGES 7,

Show

12.

be can

and

that

expressed

of

the

the

infinite

1891,]

series

sum

in

the

form

that

hence prove

[OXFORD,

1887.]

VI.

CHAPTER

STANDARD

SUNDRY

doc

f

60,i.

I.

When

a

where

of

Integration

Case

METHODS,

R

a

Positive.

is

positive

we

write

may

this

a

we

may

dx I

Q p

____.^=._.=_..___^_^==i

aJ 7/

a

If

I __

.

_

__

x/"J

bz-ac

"\2

as

as

arrange dx

If

integral

dx

If

which

ax2+2bx+c.

=

-y=

,

+-

according of

form

62 is

as

the

greater

integral +

ax

E.

T.

~

b

1

C.

as

ax .

or

"2 is

"

"

or

smh "7^

Va

ac

"

and

ac,

the

real

(Art. 36)

*

Vo2

according

than

is therefore

*

^

less

.,

,

cosh

=

or

, ~

b

+

T 1 ,

x/ac

, "

62

INTEGRAL

66

In either

the

case

CALCULUS.

be written

integralmay

in the

form logarithmic

^

the constant

"

*Ja*Jax2+ 2bx+c),

b+

log(ax+

~T=

_

62

logv

T=

*J a Also since

cosh

lz

~

beingomitted,

ac

~

=

sinh

l\/z2

~

1

"

,

sinh

and 1

,

cosh

=

I

and

.

.

1

-1" "

=

"

7--

\/ac

V^

b2

"

l\/z2 + 1

~

"7^

,

cosh

,

\/aR

,

sinh

"=.

+ b

ax

, "

cosh 1

= .

,

sinh -7=.

=

+ b

ax

, l

~

lz

~

"

x

x/aJi

T 1 -

-?

7

\/a

\/ac

"

b2

forms therefore may be taken when a is positive b2 is greateror less than ac respectively,

which and

Case

61.

If in the

write

a=

"

II.

Negative.

a

integral A.

Then r

ZJ

or

"7=:

sin

'

"

/

)*Jax2+2bx+c

1

or

dx

f

our

.

integral may dx

a

be

negative

be written

INTEGRAL

68

Ex.

dx

(

Integrate

2.

CALCULUS.

J This

integralmay

written

be

dx

I

therefore

and

is

"

sin"1-^^"

=

.

\/2 which

also be

may

\/41

expressed as -^cos

-F="

*/41

V2

EXAMPLES. 1.

Integrate

{--^" JV^

2.

62.

dx

dx -,

J

A/2"Ja + Zbx+cx*dx

3.

Integrate

4.

Integrate /\/a +

\

of

Functions

J

+ 2a? + 3

/"

Integrate

dx

{

2"#"

the

3.* -2#2

(c positive).

cyPdx

(cpositive).

-.

Form

-"=====

x/a^2+26^+c be

integratedby

which, or

may

first

be done

as

putting Ax+B

in Art.

by equating coefficients Ax+B ex/

;

we

into the

may

iorm

57, either by inspection obtain

of integral

The

METHODS.

STANDARD

SUNDRY

69

the first fraction is

A

that

and

of the second

been

has

discussed

in Articles

60, 61.

EXAMPLES.

Integrate

2.37+ 3

-

x+b

POWERS 63. Sine Index.

PRODUCTS

AND

Cosine

or

SINES

OF

Positive

with

Any odd positivepower of immediately thus : integrated

sine

a

COSINES.

AND

or

Odd

Integral

cosine

"

To

integrate Isin2n+1# dx, let .'.

smxdx=

cos

x

=

c,

"dc,

Hence

fsin^+^cfo ((I-c2) dc =

-

__

can

be

CALCULUS.

INTEGRAL

70

Similarly, puttingsince we

therefore

s, and

=

cosxdx=ds,

have

Icos*n+lx dx

(1

=

s2)nds

"

L_

nn

64. Product

sin^

of form

cos?#, p

1 \7l.

/

I

'

""

q odd.

or

product of the form method admits of immediate by the same integration either p or q is a positive odd integer, whenever whatever Similarly,any

the other be.

.example,to integrate/sin5#

For

therefore

/cos%

Hence

sin5^?dx

" "

/^

sin5^ cos3# dx

cos9^? 9cos7^;

of tan

in terms For p + q

put tan =

"

x x

=

or

cot

we

"T"'

proceedthus

"

negative

a

even

of immediate

integer,the integration

x.

t,and therefore sec2^ dx

=

dt}and let

2n, n beingintegral.Thus =

)n

|

~

^

8a5

Irftan^+6a5

4-n~*-(j ,

"

p + i

:

-

admits

sin*tocos% expression

J^f~

I sin^(l sin2x)d (sinx)

=

is

and

"

~5~'

p+^

cos#=c,

/c4(l c2)2dc cos5^7

65. When

dx, put

dc.

sin xdx=

-

cos4#

__

2

I-

4--

ldt

if Similarly,

METHODS.

STANDARD

SUNDRY

put

we

cot

x

c,

=

then

"

71 cosec2^ dx

"

dc,

and

\"DPxco"xdx=

a

result the

-

the former

as

same

the

arrangedin

posite op-

order. Ex.

Integratef?^"fo?.

1.

J

This may

sura

be written -

and

/

the result is therefore

It may

also be

CcosPx sPx =

J sin6^?

Ex.

1

f

I

,

-r-^-dx

the result

integratedin -

/T

o

terms

2

.

of tan

an

^

thus

:

"

tan~5#

\,,

"

x

x=-

"

-

"

-

J taii6

being the

same

as

before,

2.

/sec" (9cosec"

66. Use

of

0 d"9

=

ftan~*0rftan0"" -f tan~%=

-

f cot*0.

Multiple Angles.

of a sine or cosine,or Any positive integral power of sines and integral any product of positive powers in means cosines,can be expressedby trigonometrical series of sines or cosines of multiplesof the angle, a and then each term be integratedat once; for may f

\cos

J

siunx nx

7 dx

=

and

"

n

ifJ

cosnx

7

sin

nx

dx=

--.

n

INTEGRAL

72

f /"-t-UU"^^

r

/^^2^^r^,_

J}x x.

i

Ex.

2,

1

/ cos

.

c/Lx =:

x

cos%

Ex. 3.

CALCULUS.

Sin 2.27

X

/

.

dx=

/"

/

_

j/(|+ J cos

=

%x + J sin

"

67. It has is odd

alreadybeen

the second

more

we

Ex.

4.

Let

cos

shown

that when

the index

is necessary,

nor

for the

x

+

c

sin

x

2

cos

=y

2t sin

x ^

in

of

case

x

x

;

then w

=

=

y

"

-,

2i sin

nx

"

yn

"

"

yn

Thus

2

cos

8^-

16

cos

6# + 56 cos4o?-

112

The be

of

sin^cos?#, where

y

=

in

"

odd.

are

q

sm

sin

Integrate I8m9xdx. x

thus

different form. will therefore discussing

result

now

are

=

-

value especial

neither p

g1^sin Ax.

/(1 sin2.2?)a? sin x

=

presents the

method

2# +

example / cos3# dx

which

J cos 4dc)dx

2# +

transformation

such

no

""**

"4"

cos

2^+70.

l(cos 8x

sin8.??

Thus

=

METHODS.

STANDARD

SUNDRY

8

-

6# + 28

cos

4#

cos

73 56

-

35),

2# +

cos

2

f

andj

"

K

irsiii8#

j

\$n$xdx="

2i L_

J

Ex.

2

=

and

cos

8#

8

"

.

"

"

-

Kcsin2^?+35#

-56

.

"

Q*

"1

"

4

o

2^

,

_J

; then

6^+8

cos

7J

sin6^cos2^="

OQsin4#

+ 28

"

/ sin6a7cos2^ o?^.

.#+t sin ,#=y

cos

Osin6^

-8"

"

8

Integrate

5.

Put

-

-

'8# + 4

"cos

8

4#+

cos

cos

6^

"

2^;

cos

4

cos

kx

10,

"

4

"

cos

2^7 +5

V,

whence

68. NOTE.

It is convenient for such examples to remember that the several Coefficients may sets of Binomial be quickly in the reproduced followingscheme :" 1 1

1 121

1331 14641 1

5

10

10

5

1

1

6

15

20

15

6

1

1

7

21

35

35

21

7

1

1

8

28

56

70

56

28

8

1

etc., each

number

being formed above

the 7th

row

0+1

we

=

1,

it and have 1 +

5=6,

at

the

once

as

the

precedingone.

5 + 10

=

15,

10 +

sum

of the Thus in

10=20,

one

mediatel im-

forming

etc.;

CALGUL

INTEGRAL

74

US.

/

and

in

occurring above and

such

multiplying out

onlyneed

we

all the work

(1 t)Q

coefficients of

(1-/)6(1 + 0

6 + 15

-

(1

will discover

the several coefficients

the

are

"

6 +

1,

5 + 9-5

+

-

1+4

+ 6 + 4 +

1+3

+

coefficients here

1-0=1,

O2

"

+

law

same

by

c,

were

6-4

4-1=3,

=

1,

required.

thus

:

"

etc.

4-6=-2,

2,

EXAMPLES. 1.

Integrate odd

doing those with 2. Integrate

3.

indices in two

ir

/

4. Evaluate

ir

r"

sin^ctr,

*0

5.

ways.

Integrate ft

Integrate sin

6. Show

r

/ cos5^?c?^, ^0

0

si 2.# cos2.r,

that

/sin x 7. Show /"

/

sin 2.# sin 3#

dx=-"\

cos

2#

-

""$cos

4# +

^\ cos

6#.

that N

f

7

.

(i.)I sm

wia?

cos

w^

a^7

=

"

cos(m+?iV v ! "

"

"

as

forming per-

the required,

2-2-3-1, + 2 +

1,

c+c?,etc.

1,

formed

are

6,6 +

+

1,

cZ2+cfa3+... by l + ",

+

(1+04(1

the coefficients

are

row

4+4-4

of this

reason

a

a,

1+2-1-4-1 the last

5-

of a+fo multiplication

Similarlyif the coefficients in work appearingwould be

The

-

+ t)2 (1 t)G(l

-

the actual

and

15

20+

-

9-

1-5+

are

student

The

in which

coefficients of

1

+ (;*/

)

-

4-10 are + 1)* 1-4+ 4+ + t)G(l figuresbeing formed accordingto the

of

before.

1

are

-

coefficients of row

the

1\6/ "

appearingwill be

coefficients of

each

(y

product as

a

^

cos(m 72-V y )", "

"

.

INTEGRAL

76 and

CALCULUS.

generally s"5

^3

2w+2ajdx where

c

70. Odd cosecant

By

=

can

c

-

cot

=

"c"

-

WC"-

-

"

have

we

.

.

.

x.

of

positiveintegral powers thus : be integrated

differentiation

/"2n-fl ~

at

a

secant

or

"

once

d "

and

(n + l)cosecn+2" n

cosec7lo?

"

=

x cosecna -7-(cot

"

doc

whence 1 ) secw+2^jdx

(n +

tan

=

x

secn#

+

cosecnx

+

^

Ise

and + (ti

1 ) cose.cn+2xdx =~coix f

Thus

as

x

Icosec

and we

/

sec

infer at

may

dx

c?x

#

cosecn#

n

dx

I

\

=

+ ^V logtanf^

=

logtan^ ,

of integrals

the

once

sec3#,sec5#,sec7cc, .

.

.

by successively puttingn

; =

cosec3aj, cosec5#,etc., I, 3, 5, etc.,in the above

formulae. Thus

/sec3#

dx

^ logtan f

J tan

x

sec

/sec5^?da7= J tan

#

sec3^+ 1 /sec3^

J tan

x

sec3#+f tan

=

=

x

+

^7 sec

etc.

^V

+

-

x

+

f logtanf

-+-

),

SUNDRY

formulae

71. Such

METHODS.

STANDARD

A

as

student

77

called

are

"

"

REDUCTION

with

will meet

others

formulae, and

the

in

postpone till that chapter the of the integrationof such an expression been have as except for such cases

Chapter

VII.

consideration sin^cos^

as

many

We

alreadyconsidered. 72. Since is

a

a

of or

a

cosine

a

cosine

73. may

that

POWER

OF

tannx

dx

=

=

TANGENT

tann

and we

#

cfc

Itan%c dx

tangent

a

~

=

COTANGENT.

we

have

/tan3#cfo?=

cotangent

or

2x(sec?x l)dx "

f,

"

logsec

,

tan?l-2#cfe.

J

1 #,

(sec2# l)dx "

=

tan

integratetan3#, tan4^, tan5#,etc.

may Thus

==

a

able

now

"

n

Itan

are

Itann-2a3c?tan# ltanw-

-^

since

of

power

negative power

OR

idM.n~lx

And

we

positive

a

cosecant.

of Any integralpower be readilyintegrated.

For

negative

a

cosecant

or

sine, and

integralpositiveor

of a sine, cosine) secant, or

INTEGRAL

is

sine

or

secant

a

or

it will appear

cosecant

integrateany

to

of

negative power

power secant

of

positivepower

/tan x(sec2x-l)dx

x

"

x,

78

INTEGRAL

CALCULUS.

[

f

2

3

By continuingthis

process

shall

we

evidentlyobtain

_

2?i-l

2^-3 # + (-l)nff, (-l)w~1tan

+

tan^^

tan-+^=^^

and

t

_

Similarly Icoinx dx

cotn

=

~

2

cot71-1^ =

"

~ --

r-

J

71"1

whilst

icot^^aj

=

cot2^ ^

=

and and

therefore

we

f |COtn-2#CfcE,

logsin x. I(cosec2^ 1 )dx

may

"

thus

admits

"

integralpower of a tangent of immediate integration. f

Integrationof

We

may

write

a

+ b

x

"

#

or

cotangent

dx

etc. \a+bcosx, cos

x

;

etc.

any

74.

cot

integrate

cot3#3 cot4#, cot5^, Hence

=

as

j, s2|sin2| -

METHODS.

STANDARD

SUNDRY

79

(a

or

(fa

2

Thus

-AgU-j

=

"

6

"(1)

or

CASE

I,

If

a

"

b this becomes

tani a-

6

/a+6

,

tan

or

?| 2J-

Since

we

may

write

this

as

"

b,

1-

1

^ =5- COS

"

g

+ b

2

INTEGRAL

80

CALCULUS.

1 or

COS"

"

7

*-"

+ bcosx

CASE

II.

If

a

in the form b,writingthe integral

"

(K

dian.~ ,

(2)

in

placeof

the form

UjiAj

A J fti _1_ + bcosx f*na

A b

by

case

Art. 54

J.

"

sv*

in this

have

(1)we

n a

"

IT. 1 Ib +

"

17 a

"

Ib+

Vf^ v6 +

,

a

x

V6^~tan2 a

+

a

"

\/b

"

a

tan

a

tan

"=

"log '

.

rjr"

.

\J~b+

b

v

"

^ "

By

Art. 33 this may

be written

tanh~:

/62-a2

or,

we

since

2 tanh

~

lz

=

cosh

~

1

1

may

stillfurther

02'

"

exhibit the result b 1

"

a.

b+

a

1

1

"

^"-

b+

or

nX

tan2^

-L-^

:cosh~3

as

a

2

STANDARD

SUNDRY

We

METHODS.

81

therefore have

b,

"

x

i.e.

dx bcosx

a+

er "

or

=

Jl?-

cosh-

a+bcosx of the real is used.

but These forms are all equivalent, forms is to be chosen when the formula

75. The

integralof

b.

"

one

may

"

-r

a+

b

cos

x

+

c sin

be im-

x

mediatelydeduced, for b cosas-fc

smx

\/b2-{-c2cos(x tan~V ), b/

=

"

\

the proper form of the integral at can be written down in each of the cases a greateror

therefore

and once

less than

^/5*+c*. dx

Ex. 13 4- 3

cos

x

in H- 4 sin

=f [ J

x

dx --

1

/132 -

52

13 + 5

_

12 1 or

$. I. C.

-i/2

.'T

"

a\

(where

:-a) coe(# a) -

tana

=

^)

CALCULUS.

INTEGRAL

82

dx

f 76. The

integral I

.

7

Ja + 6sm# ,

be

may

--

"

easily deduced

by putting

dx

f

then

"

j". o sin

Ja + therefore

and the

by

Ja +

x

its value

dv

"^o

-

cos

,

y

be written

may

down

in both

a^b.

cases

Of

f I-

=B

"

course

be

it may

also independently investigated

first writinga + b sin

x

as

+ 26 sin | + sin2|j a(cos2| |, cos

+ 26 tan cos2^(

or

a

The

then integral

-

+

a

J. tan2^

becomes

2

and

two

cases

77, The

arise

as

integral

before.

x

I" ,

may

,"

treated. dx

f

dx

be

similarly

CALCULUS.

INTEGRAL

84

certain

that,with

4. Prove

limitations

on

the values

of the

involved

constants

/"%==L=

-

J J(a-x)(x-(3) P .-(3

____

and

/

integrate

(x

v

a)(/3 x)dx. "

"

a

5.

Integrate ,.

JSC

("'") /"""

dx

C

v

"}

.

.r

\ (1""

Ou27 cos

a

+

f

}3(l-s

(v-)

.

U11'-'

J

r

"^%

r

\

,

'

/

*

\

^'''^

COS.T'

-

2^2 r

4- cos

and

+ cos

a

x

(viii-) prove dk

Integrate(i.)f(ii.)/

^ V

C?Jt' ____

a(^

-

f

b)+

V

6(^ a) -

"**

(iii.) J

7.

Integrate 7f I

8.

Integrate

f- ^ -

J sm^

9.

Integrate

-

.

+ sm2^

fcos201ogcos^+shl fa. COS0-S1TL0

J

10. Interate

1+cosx

.77

J"2siii2"9 + 62cos2^'

o

6.

+ sin

^^

(vii.) cos

a?

11.

Integrate

12.

Integrate /

_dx.

+ sin

x

sec^

Integrate /

dx.

"

J 1+

14.

x

sm

J VI

13,

METHODS.

STANDARD

SUNDRY

x

cosec

Integrate /-"f^fl_f_. J

v

b tan2#

+

a

fVr^

15. Evaluate

"^'

"

/ 1 +

x

sin

o

16.

Integrate [****"****"",. J logtan ^7

17.

Integrate

.

Vsin 2(9

fcot0-3cot30

18.

IntegrateJ

19.

Integrate / J Wo?

20.

Integrate /"7

~

J (x si

21.

Integrate f-^

22.

Integrate fA/_ V

'

*

CQS

~

cos

+ cos (9(1

1

sin

^ 6y

6")

+ cos 6")(2

f

23.

24.

25.

Integrate Integrate f^ (sin0 + Integrate f J

.

" .

"

^?

2

"

sin

a?

"mg-coeg cos

85

CALCULUS.

INTEGRAL

sin"1

I

Integrate

26.

dx. -

.

l+x2

J

27.

?

Integrate J

\"

( sin^, 28.

{^^dx, and

that prove

J

'

sin

sin"r

[*"-X-dx,

Integrate 2#

J

sin

3^

J

sin

4^7

5 +

[THIN.

COLL.,

1892,]

CHAPTER

VII.

REDUCTION

FORMULAE.

FORMULAE.

REDUCTION

Many

functions

immediately

reducible

79.

forms, In

and

some

cases,

connected

of

whose

by

another

to

the

For

be

itself

form

in

the

Such

methods

terms

be

formula itself

any

rate

of

integral

that

shown

of

terms

may

with

standard

obtainable. be

linearly

the

integral

be

may

easier

either

mediately im-

integrate

to

of

(a2 + #2)^fe

J(a2+ #2)^fe,and

which J(a2 + ar)^cfe,

I(a2+ #2)^cfcmay

connecting

Reduction

80.

it will

in

at

the

not

are

function.

original

instance

integrals

which

or

of

directly

not

are

algebraic

some

integrals

other

or

sucft

however,

expression,

expressed

one

integrals

integrable than

whose

occur

being

be

relations

algebraical

this

can

latter

standard

a

inferred.

are

called

Formulae.

The

student

have

will

already

realise been

that

used.

several

For

reduction instance

the

CALCULUS.

INTEGRAL

88

parts of Chapter IV., and It is proposed to consider and such formulae more fullyin the presentchapter, of some for the reproduction to give a ready method of the more important, of Integration method by the formulae A of Art. 70.

where

81. On the integrationof xm-lX* for anything of the form a+bxn. In

several

the

cases

be

can integration

X

stands

performed

directly. I. If p be

the positive integer,

a

expandsinto

binomial

finite series, and each term

a

in

is integrable.

/v"

Next and

s

p fractional

suppose

=

-,

r

and

8

beingintegers

positive. 777/

II. Consider

the

Let

X bnxn~ldx

.'.

when

case

=

=

a

"

+ bxn

is =

szs~ldz zs~l

\x

r-

J

and

when

is

"

bn)

a

this expressionis integer, positive

directlyintegrable by expanding each term. integrating III. When

"

positive integer.

zs,

f

and

a

is

a

the

binomial

and

the expression negativeinteger,

(zs-a)~n+'

FORMULAE.

REDUCTION

may

be

may

then

partialfractions,and the integration proceededwith (Art.58).

into

put be TD

If

IV.

r

H

"

is

"

proceedthus

may

89

we integerpositiveor negative,

an

: "

rn -

, _

rn

m-\ and

by

is

either

III. this is integrable when

II. and

cases

positive or

a

substitution

777

or negative. integral, positive,

S

n

Three

thereforeadmit of integration by simple substitution.

cases or

(1) p (2)

positiveinteger.

a

"

an

integer.

[-p

an

777

(3)

Ex.

Here

"

integer.

Integrate (^(c

1.

m=6,

n

=

3, and

=an

"

integer.

n

Let so

that

Then

the

"

r

\-- is

"

"

negative integer by the That is, the expression is

a

b + ax~n=-zs.

integrablewhen

-

--

%x*dx=

integralbecomes

2zdz.

mediately im-

INTEGRAL

90

Ex.

Integrate/ x*(a?+ x^dx.

2.

Here

CALCULUS.

m

=

", n

3, p=b

=

and

is

+p

"

integerc

an

'

n

is / integral

The Let then

-3-. XT

the

and

becomes integral 9 *

might be put

which

6, the

process of will be avoided effected (Art.70). =

sec

formulae

82. Reduction + 6xH

Leta with

xm

then

^C;

=

of the

any

fractions. If,however, z be put partial tions puttingthe expressioninto partialfracand the final integration may be quickly

into

\xm~lX^dxcan

to according

Let indices

P

=

n

-

-

1^

the

and

m+n

lX?+ldx,

-

Xm+n

rule following

"X+1JTya+1 where

of x

connected

"

\xm-n-IXPdx, xm

be

six integrals : following \XP

-

\xm~\a

for

X

A

-

~

1

: "

and

JUL are

in the respectively

two

the smaller

expressions dP

whose

are integrals

Find

-p.

arrange Re-

linear functionof the expressions and to be connected. are Integrate, integrals this

whose

to be connected.

the connection

as

a

is complete.

CALCULUS.

INTEGRAL

92

Integrating, P=(n

/"( 1) I(x2+a?fdx-na?

+

and n+I

Putting?i

=

5 and

ft

3,

=

(( J

and Then

3i

6.4

Ex.

3.

Calculate

the value

of

[^x^-^ax-x^dx,

shall endeavour

positive integer. We

m

being

to connect

with fxm~l*J%ax-x*dxy l%m"JZax-x'*dx

(xm~\2a-x*fdx. with (xmJf^(Za-x?dx

i.e.

Let

P=^m+1?(2a-^)1r accordingto

the

rule,then

Hence

(m

+

xfdx 2) fxm+^(2a -

-

xm^(2a^ -

+

(2m + l)a fxm~\2axfdx -

a

FORMULAE.

REDUCTION

a

93

.

xm*J?Ltix o

ra

Jo

+ 2

+ 2

m

o

/la, .

_

xm*j%ax

x*dX) and

-

be

m

2ra-l

2m

to find

Now

1

+

IQ or

+ 1

5

"

dx

=

sin (

a

sin 0.

^l^ax

Also

when

#=0,

when

#

"

Hence

70=

Hence

/

x^

"

=

fVsinW^-

2a,

0).

we

have

$=0,

we

have

O

T(l -

1)...3

-m-

(m+2)(m

+

m '

3

"cos

a

d

an

3

x^dx, put

x=a(\ Then

.

etc.

_ "

'

"*4

m

fj^ax

I

3/ J.m-z

2m-3

-1 '

m

2

m

'

"mT2

a

-

.

--

-

m

-3

2m

2m-l .

positive integer,

a

l)...3

cos

+2?r_ 2

=

TT.

20)rf0

(2m + l)! m!(m + 2)!

EXAMPLES,

Apply reduction 1.

/ J

2.

the

rule formulae

stated

(when

in Art. 82 to X=a + bxn): "

obtain

the

following

INTEGRAL

94

3.

CALCULUS

(,-^ J

4.

( J ^P

=xm^ (a"*-*X*dx

.

{x"+n

mm]

J 6.

-

/

.

Integrateout 7. Obtain m

l, m=2,

=

of

integralsof

the m

3, and

=

are integration

values

when

cases

the limits

2a.

formulae

similar rule may

be

for

sin^a? co"x

givenfor

Isiupx

for

the

"

their numerical

0 and

83. Reduction A

/xm^(^Laxx^dx for

cosqx

a

dx. formula

reduction

dx,

j

be expression may six integrals : following This

with

connected

of the

any

"

I sin^

"

\sinpx

Isin^ by

the

Put smaller two

-

dx,

\si

^x dx,

\si

2# cos?#

cos9'

~

rule. following sinX+1#cosAA+1" P where =

indices of since and

whose expressions

Find

sin^+2^ cos^

2x cosv+2x dx,

-T-,

and

cos#

are integrals

rearrange

as

a

-

and

X

*x

dx,

^

are

in respectively

the to be connected.

linear

functionof the

ax

whose expressions

are integrals

the

to be connected.

FORMULAE.

REDUCTION

the connection

Integrateand Ex.

Connect

the

95

is effected.

integrals

/" P=s

Let

=(p (p

"

=

[Note the

"

l)smp~2xcosg^(l sin2#) (q-f l)si cos9# I )sin^~2# (p -h ^)sin^ cosPx "

"

last two

lines of rearrangement

*

Hence

.

P=

(p

/sin.^

I ) /siii^~2^7 cos9^

-

8m

cos%^

=

dx

(p

-

*~^X cosq+l*

-

+

where

or

q

is

q)Isi

P + qJ

in the

integral fsin*4?cos?#e"i? with 1

.

/

smp+2x

cosq.v dx.

2.

/siii^

3.

I smpx

4.

/*sin^-

5.

/ sin^

+

in

the

case

integerthe complete immediately[Arts.64, 67].

EXAMPLES. the

functionof

odd

an

integrationcan be effected The present method is useful integers. q are both even

Connect

linear

however, that

remembered,

either p

a

+"zi (*

p + q

It will be

as

sin^~2^7 cos%],

and

sin^cos^

.

"

cos"~23? dx.

2# cos"~2# dx.

case

where

p and

INTEGRAL

96

6. Prove

CALCULUS.

cousin-1*

fsin^^

that

J

n

this formula

Employ 7. Establish

f^,.

"-I

_

to

formula

a

J

n

integratesin%, sin6#,sin8^.

of reduction

/cosw#

for

dx,

Integratesin4

8.

calculate the

84. To

integrals

V

71

f 2"

.

I smn#?

5^n =

and

aa?

fl I

(7n =

J

J

0

Isinn^cdx

Connect Let P

=

0

Isinn

with

sinn~3#cos;E

~

2x dx.

the rule; then

accordingto

dP sunx

"

"

"

"

dx =

(n

l )sinn2x ~

"

"

smn~lxcosx

f lsinn^a^= J

n

.

.*.

x

J,

we

=

sin" -^

71"1

be

even

cos

^

"

---

71

71

n

vanishes

"

x

=

this Ti-l

71"2

3 ^

71

"

5 a

---

*

^

7i

"

Ti-3

"~'ii^V"g

5

""w-

4

to

ultimatelycomes

when

0, and

have

=

if

r

2, when

less than

integernot

.

-\smn~zxdx. J

"

n

since

Hence

If

"

, ---

3

Iff

4

2j J

?"

is

an

also when

FORMULAE.

REDUCTION

7i-3

Ti-1

3

1

TT

that is n-2

n

If

n

be odd

1

"

il/

Ufl

"

-L

^^

ll/

^^

Q

A,

O

TJ

9 Z"

" "

.

"

"r"

71

and

422'

similarly get

we

f-j

.v

'"

n

2

"

I sin xdx

since

97

=

*

*

\

*

rl w U**/j

dm w E*-1-11 w

"S I

5

-|

.

I

"P

3J

cos

"

/" 2"

I

r

x

1

=

o

7i-3

Ti-1 we

have

$n

=

1

In

a

4

2

-

similar way

"

2'"

5* 3* be

it may

that

seen

I"cosnxclxhas o

the precisely

in each value as the above integral This may be shown too from odd, n even. case, n other considerations. These formulae are useful to write down quickly of the above form. any integral same

/""".""",, "^|."

[The student should notice that these are easilyby beginning with the denominator. of natural numbers ordinarysequence Thus the first of these examplesis (10 under

with

most

We then have the written backwards.

is

(3 under

odd,

in

2) and

factor

forming such no

factor

-

35 E. I. C.

down

9)x (8 under 7)x (6 under 5),etc.,

stoppingat (2 under 1),and writinga first denominator

written

G

is

-.

a

But

when

sequence

written.]

the

it terminates

INTEGRAL

98

CALCULUS.

r"

85.

To

investigate

formula

a

2si

for 0

integralbe

Let this

denoted

by f(p,q) ;

then

since

tanP^^ J

p + qJ

p + q q be

have, if p and

we

less than

GASP:

and positiveintegers,

p

2

I.

If "" 6e

ei"e7i

2m, and

=

# afeo

(2m-l)(2m-3)

even

=

3)...l

m

"/v

^/^

"

9""

\

/(O,2")

/i

7/1

^i

"

1

2ft

2n,

=

)

/(

2rv

andj

be not

"

3

1

v

2

TT

=

0

Thus

CASE

II.

If p

6e

=2m,

6^67^

2m~1

q odd

-/(2m-2, 2^-1)

/(2m, 2^-1)=

' -

and

and

=

=277 etc.

"

1,

INTEGRAL

100

These

CALCULUS.

will

relations

n-{- 1 is either

T(n + 1) where

2k

being For

1

+

'

positive integer.

a

instance,

T(6)

=5F(5)= =

This not

5 .

4F(4)

5.4.3.2.ir(l)

V-) =F(f

do

to

an

2

k

sufficientlydefine integer or of the form

found

be

)= S

.

=

.

=

is

called

propose

to

enter

.

3r(3)

2F(2)

5 .4.3.

=

5!

PXiHf

function

4

5

I- fr(f )= i .

.

a

Gamma

into

.

|

.

f

.

f r(" )

function, but

its

properties

we

further

here. The

products

1.3.5...

2n-I

2.4.6

2u ...

TT

which

in

occur

the

foregoing

I

of

cases

sin^0

cos?0

o

be

may ^

so

expressed

at

once

2n-3

^(2n+l\_2n-l 1

\~~2~)

in

of

terms

this

lr/l\

2n-5 '

2

2

2~

2

that /y

2

and

sothat

Hence

in

Case

function.

I.

7T

V2/'

d9

FORMULAE.

REDUCTION

In Case

101

II.

In Case III.

we

evidentlyhave

the

result.

same

In Case IV.

It will be noticed therefore have the same result,viz.,

that in every

case

we

7T

f and

the

that the ^

sum

This

of the is

?9 1

+ l

and

#4-1

^"

the

convenient of the above quicklyintegrals a

the denominator

occurringin

+1

very

is

in the numerator.

formula

for

evaluating

form.

IT

Thus

rs f \in"6" cos80 dO

=

-*

.,

__f f j-V^TTj" f f i|^/?T "

'

"

"

*

"

_

5?T

2.7-.6.5.4.3.2.f~~215'

CALCULUS.

INTEGRAL

102

87. The student been pointed out

should,however, observe (asit has

that when either p or q previously), both of them odd the or are expression integers, without reduction sinP$cos?# is directly a integrable formula For

at all.

instance,

[sin^(l-sin26'Xsm6'=^7 (sin66"cosW6" 79 J =

J

and

Similarly, -2 0 cos26"(l cos2"9+cos46")dcos "i

COS3"9,QCOS5"9COS76n"

"Jr4-"-*^*-

+2"

-

3

But

when

p

and

are

q

if the

or integralrequired,

other than

0 and

both

^,

must

we

.jo,!

and limits of even

either

use

the indefinite be integration the reduction

tt

formula

of Art. 83

or

proceedas EXAMPLES.

Write

down

the values

of

s

in Art. 67.

FORMULAE.

REDUCTION

the formulae

prove

(1}

I sm2mOcos2n6d0

(2)

I?L_?.-_. 2

="

J

-Em+n

f sin

J 4.

103

-Bm+n-l

Write

down

the indefinite

0 dO, fsitfO

of integrals

fsitfO cos50 dO,

cos3"9dO, fsitfO

cos

fsin70 cos20 d09

cos4^ dO. fsiu60

Evaluate rT

rf

/"

sin5^cos2^^.

/ sm26

/ sin4#"w,

__"

J

J

0

0

0

/3" 7T

7T

6.

7T

/-^

/"

/"

J

"T

J

'

0

7. Deduce

the formulae

of Art.

84

/

for

(

sm x

dx from

r("ii)r(z"l) y 7 V

result

27

V

"

f

of Art. 86.

EXAMPLES. 1. Prove

that

(a) I cos2w"" ^ "/

(b)

=

-1 tan "/" + cos-nc/"

M

^^t

\

-

~

\ fcos2

2iii/ J

the

INTEGRAL

104

a formula Investigate

2.

when

of reduction

by

7.

=

of reduction

for

that

._J_ 271+2

2

27i+ 4

tegrati in-

completethe

[ST.JOHN'S COLL., GAME., 1881.]

show integral

of this

means

to applicable

and positiveintegers,

a formula Investigate

3.

and

and are n if ra=5, 7i

m

CALCULUS.

271+ 6

2.4

2.4.6

adinf

271 + 8

2. 4. 6. ..27i

~~3. 5.7...27i + l' Sum

also the series

1

1

1

1.3.5

I m

271+1

1.3

1

I

2

2^ + 3

1 _1_

I t

,

2.4

"

27i + 5

2.4.6

.

,

\f

OjCll ITlrT

271 + 7

[MATH. TRIPOS, 1879.] that

4. Prove

2n+l

/

(rf

*-, 6. Find

reduction

formulae

prove

for

(a) x"(a+ bx)P*dx,

(y)

J

and

*^*"+a")"*r, (S) /2p+l /*"(*"

obtain the value of

7. Find

a

reduction

and positive integer,

-

.

formula

for

[COLLEGES"CAMR]

Ieaxcosnx dx, where

n

is

a

evaluate

[OXFORD, 1889.]

/#wsin from

Deduce

dx

x

the latter

a

1 05

for

of reduction

formulae

8. Find

FORMULAE,

UCTION

RED

/eaxsinnx

and

formula

dx.

of reduction

for

Jcos

8in"*"fe.

a*

[COL1KGES

%

189o.]

Tt

9. If

rT / si

un=

o

that

prove

and

^-l-

deduce

un=

-^^-+^

-"

2n+1 In

--3),

rc-

rC-f1-/ /b --

-

--

sv

+

\ f

""

"(^" l)(n-2)

n("" 1)

(2ro-lX2ft-3)...3 TT '

8'

[MATH. TRIPOS, 1878.] 10.

Show

that

1 V

/

1 \

'

wi"

2/3

n"2fifJ")

[TEIN. COLL., CAMB., 1889.] 11. Prove

that 1 2.4.6 ...2m _1.3.5 ...(2w-l) TT ~2.4.6...2m 4~3.5.7...(2m+ l) 2*

7

*

'

1

12. Find

of reduction

formula

a

for

f-~=L ^

3. 5. 7.

..(27i+l)l

where

a1} a2,

13.

Show

the binomial

"

Show

that

1

coefficients.

[ST.JOHN'S, 1886.]

that

/cos

2TO

are

...

v'^

=

mx

r+-

cosm#

dx sn^

mm

4-

"

sn^

, ~"

~~

~T72 where

m

is

an

integer.

[COLLEGES

a,

1885.]

14. Show

m

CALCULUS.

INTEGRAL

106 that

being a positive integer.

[OXFORD, 1889.]

that if

15. Prove

Im,n=

sin

I cosm#

(m + n)Im)n=

cosmx

"

da:,

nx

cos

93

92

/

[if ^-l^(2+i+i 1

If

16.

,1

prove and

that

/m?

cos

cos%^7

r

,

show

I cosw#

=

w

/m

w=

"

d f cosmx\

^

^

m2-^2

-

-

)+

m(m"l)T v m*-n2

./^m-2, n,

that

cosm^7 sin

Hence

-+^J- [BERTRAM]

cEr\oos9u;/

/I prove

9"i\

+

dx,

nx

"I

-

-

m/m_i?n_i?

+

nx

that

^m '

w=

1

--

"

+

m

find the value

?i

cosm#

dx\

wm_i

"

M_I.

+ n

m

(when

/If

7^^;

m

is

a

sin 2mx

of integer) positive

dx.

[7,1887.]

o

18. Prove

that

' r2 / cosnx

cos

nx

dx=

J 19. If

m

+

n

be even,

/

prove

IT "

T

2n+1"

[BKRTRAND.]

that

co -

m-n. i

2

1

. -

2

[COLLEGES, 1882.]

CALCULUS.

INTEGRAL

108

Find

28.

reduction

a

formula

for

the

integral

xmdx '

(log#)n' Find

29.

reduction

a

[OXFORD,

1889.]

for

formula

xmdx

C

[ft 1891.]

that

Prove

30.

if

X=a

/nZ""=[ST. reduction

Find

31.

formulae

\CLJ

(Q\

JOHN'S,

1889.]

for

I tann

x

dx.

dx

f

.

J (a+"cos#

32.

Establish

parts,

u

and and

the v

following

being suffixes

functions

+

formula of

integrations

x"

csin#)n

for and with

double

integration denoting

dashes

respect

to

x

:

by

entiation differ-

"

/ /u

+

(-

l)n-1nu^n-1hn+i

+

(-

l)"n I uMvn+idx

+

(- l)n

{dx (u^vndx. [a, 1888.]

VIII.

CHAPTER

f INTEGRALS

The

88.

of

dx

\^

FORM

OF

integration

EXAMPLES.

AND

METHODS

MISCELLANEOUS

expressions

."

of

the

form

dx

be

can

If

be

The

Let

X

and

II.

X

linear,

III.

X

quadratic,Y

and

performed,

89.

Y

I.

X

CASE

best

in

effected

readily

I.

be but

X

are

Y

both

for

cases

which

functions

linear

of

x.

quadratic. linear.

Y

both the

all

the

quadratic is

process

and

substitution

-fe

Y is

both

:

"

dx

more

linear.

integration troublesome.

can

INTEGRAL

110

CALCULUS.

Putting cdx

,

we

,

nave

=

at/.

=

-(y2

,_

and

+ b

ax

e)+ b,

"

C

and

becomes

/

21

"

the standard

forms

Jy Ex.

which, being

"

^

jay2

7 + bc .

"

ae

2_^ "A

is

2,

,

one

of

immediatelyintegrable.

f

Integrate /=

"

J (xLet

then

Thus

y+lj

y-l

90. The

same

viz., substitution, *Jy=y will suffice l(fi( T\f] 'IT

for

the

rational F

of integration

I

-^^

when

functionof x, integralalgebraic

eacfe linear.

are

Ex. .

^(cc)is

Integrate /==

f

J^-

Writing "/^+ 2=7/, we

have

%dy

and

.r

=

?/2 2, -

and

X

any

and

METHODS

MISCELLANEOUS

so

-"L ="="

that

EXAMPLES.

AND

+

24/-32/

Ill

+ 16

division).

(by common Thus

91. CASE The

II.

proper

linear,F quadratic.

X

substitution is : "

X=\

Put

y Let

Putting

ax

+ b

=

-

,

t/ we

differentiation, have, by logarithmic

dy

adx ax

and

ex2 +

ex

+ b

y

+/=

6Y + ft) -2((+/ a\ a2\/ -

-

Hence

the

has integral /=

form which

has been

been

-

/

/

reduced

: "

alreadydiscussed.

to the

known

INTEGRAL

112

Ex.

f

Integrate /=

Let

./

then

#+l=y-i,

CALCULUS.

and

__=_

#+1 /=

y

__

i+i-2

i+%-y2

V

JI*

92, It will

now

that any

appear

expressionof

the

f

form

J( be

being any integrated, "j)(x) For by algebraicfunction of x. can

"b(x) we

can

express

-

,

.

in

rational

integral division

common

"

,,

the

torm

Af

Axn+Bxn~l+

...

+Z

being the quotientand

We thus have reduced the process of a number of terms of the class integration

remainder.

M

the

to the

Eaf

and

of the class

one

f

M -dx.

and latter has been discussed in the last article, of the former class may be obtained by the integrals

The

reduction

formula

^(^_^/)4_2r-l TO

2r

r-lf

e v

c

'

v

r

c

METHODS

MISCELLANEOUS

where

F(r)stands

f

for I J

The

J

By

dx.

,

\Jcx*+ ex+f

(x+l)*/x2+l

*2+3*+5

division

exercise.

an

as

f ^2 + 3^+5-^.

Integrate /=

Ex.

113

xr

this is left

proofof

EXAMPLES.

AND

*=x

+ 2 +

-

Now

and

to

x

integrate /

we

-

and

put #+!=_

Thus

93. CASE The

III.

proper

X

quadratic,F linear.

substitution is : "

+/Y=y.

Put

T /=

Let

f

dx

I

-

-

"

J (ax2+ bx + c)V

Putting

*Jex+f=y, edx

and ax24-"^ +

E. I. C.

c

7

reduces to the form

j

"c

+/

get

INTEGRAL

114

Now

e)Ay* be thrown

can

"

-j

4

p

.

dy

-f

I becomes

and

CALCULUS.

2

partialfractions

into

n

i

as

and

each

94. It

fraction

is

is also

evident

the

that

substitution

same

of of expressions integration

for the

be made

may

by foregoingrules. integrable

the

form

*(")

f

dx"

_

J

"p(x)is

where when

^/ex+ fis put equal \

the

__

c)\/ex+f rational,integraland bx +

(ax2+

form

to yt y"

/,/2n-2_i

/j/2nj_ \

"^

algebraic ;

for

reduces

to

"

"

0 2

7

_i_ "L

which '

by divisi"n'

-

and

the

rules

for

partialfractions,may

as

and

Ex.

each

term

is at

Integrate

once

integrable.

/=

Putting \/^+ l=y,

we

have

-7====2e?y,and v^

+

1

_

2

be

expressed

INTEGRAL

116

CALCULUS.

/becomes

Thus

(a2 "

W

Also that

so

and

/ reduces

Thus

If

a

6,we

"

further

may

to

arrange

/

as

i

/"

V^TP

,

-

Ex.

2.

.

-,

,

Integrate/=

1

.

/ -

J (2x2 \/3^2-2^ 2x2 -2^+1) -2^ __

dy

3^-1

2#

+ 1

"

1

_

~~

values yj2and yf of ;/2 and minimum are given 2 by x \ and # 0, and are respectively and 1, so that for real be not greater than 2 and not less than 1. values of #, ?/2must The

=

maximum

=

MISCELLANEOUS

2^2

/ becomes

/2#"

r("g3-2ar+l )(2a72 J x(x-l)

Now

EXAMPLES.

t-tfm^l-g

and

_

AND

yi-f=^-y"='

Now

Thus

METHODS

-

2.37+1

~

1

Thus

'=/(-,== 2

=

cosh"1?/ +

2

cos"1-^-

,

1

/3^2-

N/2 \2^-

EXAMPLES.

Integrate 1.

4.

2.

5.

3.

6.

_4 "

"

117

CALCULUS.

INTEGRAL

118

96. Fractions

ina;+CCOSa;. CT+f + Pisin^ + CfiOSX

of form

al This fraction

into the form

be thrown

can

A

"

(Oj+ ^sinx A, B, G

where

A +

and

each

97.

+ c1cos x)

chosen

so

c-fos x) that

-Bc^ + Cb^b, is then integrable.

Ca^a, term

the expression Similarly a

be

may

x + (ax+ b^siu

constants

are

~

1

1

(a

+ b sin x-\- c

cos

x

arrangedas

+6^+0

.)"+

cos

_|

-

x + (Oj+ 6xsin

third fractions may

the first and reduction formula and

c-[Cosx)n

be reduced

while [Ex. 25, Ch. VII.], is immediatelyintegrable.

98. Similar a

+ b sinh

ai + 99.

remarks x

+

c

a

x

by

+ b sinh

x

+

c

cosh

x

#' (ax+ 61sinh # + qcosh x)n' x + Cjcosh frisinh

Some

Special Forms.

It is easy to show

that sin

a?

Isin^r sin

"

c\

a .

"

'sin(a 6)sin(a c) "

"

a

the second

fractions of the form

apply to

cosh

~

a),

AND

METHODS

MISCELLANEOUS

EXAMPLES.

119

sin2,^ and

"

"

7

"

"

r"

-.

-.

r

-,

T\".

a)sin(^ 6)sm(# "

c)

"

1

sin2a 1/ ^^i E51111 1

""

sin

f I

whence

-r",

r)\"nY\( rt (-t/ U lollll

/

,

" ^^

__ ^^

Iv

/^ I/ I

"

"

M

-^"7

/

/Y

tv^i

" ^^

\

.

.

-

-

7

,

r

r

/,

.

. v

.

a), lo^sm(o3 v 6 "

c)

" "

"

sin2^ dx

f

,

ollll

c?a?

#

.

Ssina 6)sm(a sm(a (a 6)sm(a and

/7* cC"

QTTn

r

-

--

-5-7

r-^-p

,

J sin(^ a)sm(x "

.

/ x o)sm(x

"

c)

"

sin2a

S

-

"

sin

;

(a

"

1\

integrate any

tan

:

/

"

7r

.

2

"

*

shown

has

a

"

"

6)sm(a c)

"

generallyHermite of the expression

100. More

x -

-

how

to

form

8,cos 9) /(sin sin($ a1)sin(0a2) sin(0 an)' f(x,y) is any homogeneousfunction of _

_

"

"

"

.

where Ti

"

.

.

#, y

of

1 dimensions.

For

by

f(t,1) (* Oj)(t a2) (t a^ -

-

-

.

.

.

_

_

-

-

-

.

.

.

,

x

^" ax

which

fractions partial 1) /(a,, (ax egCoj a3) (ax "") of

ordinaryrules

the

may

(a2 ^Xag "

a3)

"

...

(a2 an) ^" "

be written

_

__

^r((ar a1)(ar a2) "

"

...

(the factor of the above

a2

(ar" aw) ^

being omitted coefficient).

ar

"

*

ar

Proc.

Lond.

Math.

"

ar

in the denominator

Soc.,1872.

CALCULUS.

INTEGRAL

120

Putting

"

tan$, a1

=

tana1, a2

=

tana2, etc.,this

=

becomes

theorem

/(sin0, cos 6) sin($ a1)sin(0a2) sin(0 an) "

"

"

.

.

.

/(sinar, cos ar) r=isin(ar OL) sin(ar an) sin($ "

"

"

...

ar)

Thus

W: /(sin0,

9)

cos

/(sinor, ~

"

"

^ism(ar

%)

_logtan^ 2

"

^ "

a,)

cos

"

7

./

x

7

"

.

.

.

sm(ar

\

"

J-'-'ii t/ctj-j.

;-:

.

an)

EXAMPLES.

Integrate sm

cos

^

^^7

cos

cos cos

cos

0

O.

2# ^

3#

cos

2a

"

cos

a

"

cos

3a

"

cos

a

"

sn

sin

GENERAL 101. There

are

^7

"

"

"

cos

a

sin 2a sn

a

D.

.

x

x

sin 2^7

K

"

cos

-

4

#(sin2,#sin2a)' "

PROPOSITIONS.

certain

on general propositions

almost which are integration definition of integration or meaning. Thus

self evident from the from the geometrical

f(j)(x)dx= J

102. I

for each is equal to \^(^) \HC0 if "f"(x) he the differential coefficient of \fs(x).The result beingultimately ~~

*

See Hobson's

Trigonometry,page

111.

independentof z

121

immaterial whether x or plainly process of obtainingthe indefinite

it is

x

in the

is used

EXAMPLES.

AND

METHODS

MISCELLANEOUS

integral. /""

1 "p(x)dx "f)(x)dx + (j

103. II.

=

c

a

a

For if

/""

pc

of "p(x) integral the left side is \{s(b) the rightside is \^(c) is the same thing. illustrate this fact geometrically. us be the indefinite \[s(x)

"

and which Let

Let dinates

the

NJP^

N^^

be 2/ N^P^ be cc

drawn

curve

the above

Then

0(#0"anc^

=

a,

=

x

=

c,

x

equationexpresses

fact that Area

+

["j)(x)dx [$(x)dx.

104. III.

=

"

b

a

For

with

the

same

the left

and

area

the

notation hand

righthand

before

as

side is side

\/r(6)T// is [ "

"

=

^e^ ^ne or" b respectively.

the obvious

CALCULUS.

INTEGRAL

122

f"/"(x)dxf0(a x)dx

105. IV.

=

-

0

0

For if

we

put

have

we

and

if

x

=

a

dx

=

"

x

=

a,

y,

"

dy, y

0,

=

I "p(x)dx= I (f"(ay)dy

Hence

"

"

o

a

=

(by in.) fV"-2/X2/ o

=

I "{"(ax)dx (by I). "

o

in

this expresses Geometrically estimatingthe area 00' QP

the

fact that, the y and x

obvious

between

O'

Fig.9.

axes,

like take as

our

O'Q,and a curve originat 0',O'Q as

ordinate

an

our

direction positive

PQ, we may if we our F-axis,and O'X

of the X-axis.

INTEGRAL

124 since

Thus

CALCULUS.

sin"^^

/

dx

smnx

=

sm"(7r #), -

/ sinw^7dx

2

0

and

;

0

since

cos2n+1#

and

cos2n#

=

=

cos2n+1(?rx\ cos2n(7rx\ "

"

"

rir

+ 137(jfo7 COS2'l 0,

I

=

*o

ft

r

/ cos2w# dx

and

=

'0

We To

0 and

propositioninto words, thus : of the form smnxdx at equal intervals "

is to add

TT

cos2n^7dx.

0

may put such a add up all terms

between

%\

up

all such

from

terms

0

to

"

and

For the second quadrant sines are merely repetitions order. of the first quadrant sines in the reverse Or geometrically, the curve about $mnx the ordinate being symmetrical y

to double.

=

#

0

^,the

=

and

whole

between

area

0 and

TT

is double

between

that

|.

Similar

geometricalillustrations

108. VII.

If

will

apply to

other

cases.

""ti /"net

\

pa

"j)(x)dx=n\ "j

it is clear that it "j"(x), of the part consists of an infinite series of repetitions the ordinates OP0 (x 0) and JV^Pj lying between bounded (x a} and the areas by the successive ordinates and portionsof the curve, the corresponding the #-axis are all equal.

For, drawing the

curve

y

=

=

=

Thus

and

f "{"(x)dx=r'(t"(x)dx= f )dx /"a

j"wa I

I

71 1 "p(x)dx. ^"(x)aa; =

I

=

etc.

METHODS

MISCELLANEOUS

AND

EXAMPLES.

125

Thus, for instance, f2"

"

o

xdx=%

sin

F

\

J

J

"

"",

j

"

sm

T

A

=4

\276u?

-

/

Bin

J

"n

2n-3

A%n-I

7 #aa?""*4

""

I

IT

2

2

...-"

2ra- 2

2?^

-.

O

Fig. 10.

have

109. We

INTEGRALS.

DEFINITE

ELEMENTARY

SOME

the

whenever

that

seen

be performed,the can l^"(#)cfe integration

definite

the

In

many

can integral "j)(x)dx

cases,

however,

the

at

value

once

of

indefinite value

of

be inferred. the

definite

definit integralcan be inferred without performingthe inwhen it cannot and be even integration, performed. We propose to give a few elementaryillustrations. Ex.

1.

Evaluate

/=

=

{*( J

Writing we

and

have

vers~

-

a

=

TT

"

A

a

CALCULUS.

INTEGRAL

126

Hence

1=

Hence

/=

~3/2f(TTf\2ay

-

| o

Putting and

we

?/=

obtain

a(l-cos0),

f'smn+10d0

/=?an+1

7ra

=

...

accordingas

n

is

or

even

n+i

down

to

? or

l

3

22

E,

odd. ir

Ex.

2.

/ logsin #

/=

Evaluate

ofo?,

0

Let

#=--#, 2

dx

then /=

and

dy

" "

;

/ logcos y dy

"

Hence

/

#

c"

rf

rl 2/=

/ logcos

=

logsm^"ir+ / logcos

xdx

jo

\

log /I (log /f

sin

^

#

cos

c?^

o

sin 2%

"

log

o IT

r"

"j

Io88inte"fo-i

0

Put

2x=z, o?^7

then then

=

/

^dz ;

Iogsin2^^=^/logsin zdz"

I f

AND

METHODS

MISCELLANEOUS

/-^2i log 2,

27=

Thus

EXAMPLES.

/=|logl. log /?

Ex.

3.

-\

r~%

\ log cos

sin xdx"

cfo?

#

=

-.

2t

J

2i

/ -^

1=

Evaluate

log

-

o

Expanding

the

have

logarithm,we

6

If

re

/=

have

Hence

x=l

put

we

we

/

"

^-dy

"

J

/

=

"^'a?^=

/

also have e

"y, dx.

"

" "

.

6

\-x

"o

Ex.

4.

Evaluate

1

-I

^=tan(9,

Put

.-.

1=1

log(tan0+cot0X0 o

/2(log

sin ^ 4-

log cos $)6"

o IT

-

2

/logsin

0 dO

=

Tr

log 2.

127

INTEGRAL

128

110.

CALCULUS.

under

Differentiation

Integral Sign.

an

to be "p(x, Suppose the function to be integrated c) which is of independent x. containinga quantityc Suppose also that the limits a and b of the integration and of are finite quantities, independent c.

Then

will

r"

-

J0(a?, a,

For

let

u

=

f6 \ "f"(x, c)dx. a

Then

u

+ Su

=

f 0(o3,

"

which, by Taylor'stheorem,

And be

if z, say, be the

greatestvalue

of which

capable,

in the limit when vanishes and in the limit Thus diminished. ^ "

=

"

a

'

'dx.

Sc is

indefinitely

The

111. contain

c

AND

METHODS

MISCELLANEOUS

in

case

EXAMPLES.

the limits a and beyond the scope of the

which

is somewhat

129

b

also

present

volume. be used to deduce many proposition may has been performed. when one integrations This

112. new

since

Thus

L=

f --

=dx

=

(x+cyJx-a

J

-* Vc

tan-1 +

\l2=2(c+ *

a

times n have, by differentiating

we

c

+

with

^

(^+ c)(^ a)

0),

regard to

times with regard to n Also, differentiating

/-

a"

a

a,

we

c,

obtain

2

"

this differentiating Similarly, c,

we

obtain o?^?

r

IJ

latter p

"2^+1

(^+c)^+1(^-a) 2

EXAMPLES. 1. Obtain

the

: integrals following "

f (i.)f(i+*)-V*"fo. (v.)J J

(ii.)An-^-xi+ J

ar)-*^

J

*?)"*"". r#-1(2-3a?+ -(vii. (iii.)

E. T. C.

I

(vi.)

times

with

regard to

INTEGRA

130

2.

Integrate (i.) (a2+

L

62

-

CA LCUL

US.

^2)v/(a2 ^2)(^2 62)' -

-

[ST.JOHN'S, 1888.] 1

a2)^^+^

(x2 +

|UL* the values /-

sin

J (cosx /"

of

f

^

+

x

dx

a)V(cos x 4- cos fi)(cos+

cos

cos

x

y

J cotfx + a\Jcos(a;+B)coa(a: + 'v} a)\/cos(^ + ^)cos(^ + y} 4.

Prove

constants

)

cLx

I

\

[TRINITY,1888.]

6"Vacos2^ + 67iii2"9+V

sin 3. Find

I~STJ"HN'S" 1889.]

that,with involved,

certain

limitations

+ Zbx + (x-p)(ax?

the

on

d,L

C% 1890-]

"

\^ Olll

"

.

(-ap2-2bp-cft

cy*

of the

values

(x p)(b2 acf -

-

[TRINITY,188G ] 5. Prove

\(cQ$x}ndx may

that

be

the series

expressedby j\r ^.v3 "

pf p etu,

_L -"

-r

...

n-

being the coefficients of the

ND N2J Nft and n having any .

.

.

real value

expansion(1+ a) 2

,

or negative. positive

[SMITH'S PRIZE, 1876.] 6. Evaluate

the

: followingdefinite integrals "

(i\

W

fl

J

l

^2

/a(a2

+

/y.2 ^'

[ST.JOHN'S, 1888.]

^72)2

o

/"""

\

^UL'

f

** x

dx

Jo(l+tfX2+^)(3

+

7. Prove

8. Show

that

#)

[OXFORD, 1888.]

f -

that

[0x^,1888.]

CALCULUS.

INTEGRAL

132

15. Evaluate

dx

(i.)f

[I.C. S., 1887.]

o

[I.C. S.,1891.] 16. Prove

f^an^^

(i.)

J

37+

sec

J a2 a

cos

[POISSON.]

4

#

cos2.2

"

beingsupposedgreater than unity. 17. Prove

[OXFORD, 1890.]

f 1-2S"fc? g

(i.)

=

-

o

18. Prove

that =

a

"

z2)

"a3

"5

""

+

" "

-

"

-

ctf+

...

3.5.7

3.5

3

[OXFORD, 1889.] 19. Prove

that

IT

2r~% /

/-7A "^"

beingsupposed " 20.

Prove

_T

.

2 1 1

1 2 o

"

Q2

1

1

1*3-4

i

o2

2 "

^"

"

PL2 " x,6

i

1.

that

[MATH. TRIPOS, 1878.] 21. Prove

that 1

1.1

''*

22.

$(x)dx= -,"),\*a

If

[A 1888.]

F $(x)dx.

-

0

[TRIN. HALL, etc.,1886.] 23. Prove

*""6^ Jf^C~^W

that

c-a?)

^-6)

provided

b

remains

finite when

x

vanishes.

[ST.JOHN'S, 1883.]

that

Prove

24.

AND

METHODS

MISCELLANEOUS

EXAMPLES.

133

and illus/""{"##) ra$(x)dx= + ""(2a-a?)}cfci7, *

the theorem

trate

25. If

and

that

show

f(x)=f(a+x\

illustrate

26.

geometrically.

geometrically.

that

Show

\

q-pj

q-p

q-p

value of the 27. Determine by integrationthe limiting of the following series when n is indefinitely great :

sums

"

'

n

/""

+ I

+ 2

n

+ 3

(iiL)_J_+

+

n

ri

[a,1884.] n

n

n

n

x

n

*

*_+

.-+

\/2?i-l2 \/4?i-22 \/6rc-32

+

-J*/2ri*-"n? [CLARE, etc.,1882.]

(iv-) "

n

sin2/" + sin2K + sin2fc +... +sin2K" !-,K beinsr 2 J 2n 2n 2n (.

i

"

"

"

integer. 28. Show

n

an

[ST.JOHN'S, 1886.] that the limit when

n

is increased

2*

n2

3n

'2n

of indefinitely

[COLLEGES, 1892.] 29. Show

that the limit when

n

is infinite of i

i

is

Apply

/*"+*.

e^a this result to find the limit of

-('+ [CLARE, etc.,1886.]

INTEGRAL

134

Find

30.

the

CALCUL

(n\}n/n

of

value

limiting

US.

when

n

is

infinite. .

Find

31.

of

the

of

sum

value

limiting

the the

n

when

n

is

infinite

of

the

Tith

part

quantities

n+1

n '

+

2

n

3

+

n+n ~

T~J

Ti~J

"V

n

and

show

Napierian 32.

show

the

of

product the

that

If that

na

it

is

same

the

to

limiting

quantities

as

of

value 3e

:

the

8, where

logarithms. is

the

always

limiting

?ith e

is

of

the

base

[OXFOKD,

equal value

unity

to

of

the

and

n

is

the

root

indefinitely

of

1886.]

great,

product

[OXFORD.

1883.]

IX.

CHAPTER

/

ETC.

EECTIFICATION, In

113.

the

of

course

the

four

next

chapters we

of obtaining foregoingmethod limit of a summation by applicationof the process of integrationto the problems of findingthe bounded by such lengths of curved lines,the areas of solids of and volumes lines, finding surfaces to illustrate the

propose the

revolution, etc. Rules

114. As idea

for the

Tracing

shall in many cases of the shape of the we

order

of

have curve

the

a

Curve.

to form under

limits

some

rough

discussion,in of

integration, author's refer the student to the we larger may Treatise on the Differential Calculus, Chapter XII. for a full discussion of the rules of procedure. The followingrules, however, are transcribed for to

properly assign

,

convenience

of

will reference, and suffice for present requirements:"

115. 1. A curve.

I

For

glance

Cartesian will

in

most

cases

Equations.

suffice to

detect

symmetry

in

a

CALCULUS.

INTEGRAL

136

(a) If

Thus

of y occur, the curve is symmetrical powers larly Simiwith respectto the axis of x. for symmetry about the ^/-axis. odd

no

4"ax is

y2

=

symmetricalabout

the cc-axis.

(6) If

of both x and y which all the powers be even, the curve is symmetrical about axes, e.g.,the ellipse

occur

both

^ y*_ a2+62~ (c)Again,if on changingthe signsof x and y, the there remains unchanged, equationof the curve is symmetry in oppositequadrants,e.g., the hyperbolaxy a2,or the cubic x3+y3 3ax. If the curve be not symmetricalwith regard to tion either axis,consider whether any obvious transforma=

=

of coordinates 2. Notice

could make

whether

the

curve

it

so.

passes

through the

the coordinate the pointswhere it crosses coordinates present axes, or, in fact any pointswhose themselves as obviously the equationto the satisfying

origin ; also

curve.

3. Find

those parallel to the asymptotes; first, axes ; next, the obliqueones. 4. If the curve pass through the originequate to of lowest degree. These will terms the terms zero givethe tangentor tangentsat the origin. 5. Find

the

y^; and

where

dx'

find where finite;i.e.,

the

it vanishes

or

becomes

or tangent is parallel

in-

pendicul per-

to the #-axis. 6. If

of the variables,say y, in terms of the other, x, it will be in the solution, frequentlyfound that radicals occur and that the range of admissible values of x which we

can

givereal values ofloopsupon a

solve

the

equation for

one

for y is therebylimited. The existence is frequently detected thus. curve

ETC.

RECTIFICATION, 7. Sometimes reduced to the

when simplified

equationis much polarform. the

Polar

II. For

116.

It is advisable

137

Curves.

to follow

such

some

: following 1. If possible, form a table of r and 9 which satisfythe

routine

as

the

"

of 9, such

as

6

=

0,

values corresponding for chosen

curve

"^, "","f

etc.

JP

negativevalues

9 leaves the

equationunaltered, e.g.,in

a(l

=

"

both

of 9.

initial line.

r

Consider

o

there be This will be so when

2. Examine

values

,

O

and positive

of

whether

symmetry a change

about

the of signof

the carclioide

cos$).

obvious from the equation of the curve confined that the values of r or 9 are between certain limits. If such exist they should be ascertained,e.g.,if r asijm9, it is clear that r must lie in magnitude between the limits 0 and a, and the 3. It will

be frequently

=

curve

lie wholly within

4. Examine

rectilinear

or

the circle r a. has any the curve =

whether

asymptotes,

circular. RECTIFICATION.

a

117. The process of between two curve

findingthe lengthof an arc of fication. specified pointsis called recti-

Any formula expressingthe differential coefficient of s proved in the differential calculus gives rise at once by integrationto a formula in the integral calculus for findings. add a list of the most We common. (The references are to the author's Diff. Gale, for Beginners.} 118.

values

In each of the

the limits of integration are independentvariable correspondingto case

the

INTEGRAL

138

the two pointswhich is sought.

Formula

in

"he Diff. Calc.

CALCULUS.

the

terminate

Formula

in the Int. Calc.

whose

arc

Reference.

Observations.

For

P. 98.

length

Cartesian

Equa

tions of form

For Cartesian Equa tions of form

P. 98.

*=/(*/)"

ds _

dt

l(dx\*Idy

P. 103.

For Polar of form

Equation

P. 103.

For Polar of form

Equation

P. 100.

For

M(di)+(Tt

is

rdr

Pp. 103,

For

case

when

curvi

given as use

when

Peda

Equation is given

dr

105.

ds

P. 148.

For

use

when

Tan

Pola gential Equation is given

119. We Ex.

1.

add illustrative

Find

extendingfrom y="

,

yi="

the

length of

the vertex and ,

to

examples: "

the

one

the limits

arc

of the

extremityof are

#=0

and

parabolaxL=kay the latus-rectum. x"%a.

Hence

Ex. p

Find

4.

and

the

between

rsina

=

CALCULUS.

INTEGRAL

140

length of the the points at

arc

f 2- -^r

=

J Ex.

Find

5.

whose

s

where

^

of the

arc

In

\^2are the values respectively.

case

involute

of

at the

beginning

of

closed

a

be is

end

oval, the originbeing within that

the

length of

I pd\fs,for

the

the

that

exceeds

by

the limits

the

portion

perimeter of of its

"

taken.

are

ellipseof

an

of

length that

a

small

circle

tricity eccen-

having

the

[7, 1889.]

area.

p2 ^

and

"

Show

Here

circle,

Curve.

given by

disappearswhen

e

^

observed

\~dj)~~]

Ex.

a

formula

contour

-jy

of

Closed

for

it may

curve,

arc

any

=

using the

whole

=

is the

p

), c^cos2^+ 62sin2^ a2(l e2sin2^ =

angle =

which

makes

p

-

with

the

major

axis.

.A a(I i^sin2^-e*sui*\ls. -

-

\

Hence

of the

are

=

Formula

in the

Hence

vectores

=r

and

120.

where

radii

VV2-r2sin2a

length of

the

equation isp

Here

same

equiangularspiral

the

which

r2.

Here

the

of the

arc

2i

o

/

I \-^-\\f}(very approximately) ,=4a{|-lA

ETC.

RECTIFICATION,

(r)of

radius

The

circle of the

a

is

area

same

141

given by

vZ^ab^a^l-erf, /

its circumference

and

-e2

"

*.

"

4

32

circle

ellipsecircumf

Circumf.

-

).

e4

"

...

\

.

\

o

I

1 27ra(

=

/

( \lo

=

"

.

\irae* =

-

2ira

""

. _

3'2

t"4

/

4

o '

of circle], far as [circ.

= "

involvinge4.

terms

as

64

EXAMPLES.

by integrationthe length of the arc a2,interceptedbetween the pointswhere

of the

1. Find =

2. Show

from

that

the

in

the vertex

catenary y

(where #=0) =

c

smh

cosh

the

-

length of

arc

point is given by

to any .

s

c

=

cos

x"a

circle and a

x

i

-.

c

3. In

v

that

the

the

where

evolute of length of the the

it meets

4. Show

that the

viz.,4(# 2a)3 27a#2,show parabola, from its curve cusp (# 2a) to the point parabolais 2a(3v3 1). a

=

-

=

"

lengthof

the

of the

arc

cycloid,

.r=a(0+ sin 0) ^ a(l-cos"9)J y =

between 5. Show

the

pointsfor

that in the

0=0

which

and

for epicycloid

y=(a

6)sin0

+

0=2^,

is

s

=

which

b sin

-

=

26

a

5

measured beiijg

from

the

point at

measured

"=--, from

a

0=7rb/a.

222

n

When

which

show

that

cusp which

4r+y*"a*j

lies

on

the

and

s3 oc y-axis,

that x*.

if

5

be

INTEGRAL

142 6. Show

CALCULUS.

that in the ellipse # be expressedas

may

7. Find

the

(i.)r

length of =

r (ii.)

=

of the

arc

any

cos

a

=

$, j/

6sin^, the perimeter

=

curves

acos0.

r (iii.)

a6.

aem0.

(iv.)r

asin2-.

=

=

2t

Apply

8.

whose

s=_"

the formula

equationis r=a(l

9. Two

radii vectores

-f cos

be

found

that the

lengthof

in finite terms

the

curve

initial line

is act, where

of the

Find

an

in the

arc

of the

cases

when

length of the arc between (c2 a2)p2=c2(r2 a2).

the

"

that

the circular

is the

a

"

two

Times.']

yn=xm+n

curve

or

"

*m

curve

12. Find

; prove

[ASPARAGUS, Educ.

integer. 11.

cardioide

[TRINITY,1888.}

of the

OP, OQ

drawn

10. Show

the rectify

0).

equallyinclined to length of the interceptedarc of the anglePOQ. measure are

to \pdty

+

+

*m

-

can

is

an

*

consecutive

cusps

"

the whole

lengthof

the

loopof

3ay2=x(x-a)2i.

the

curve

[OXFORD, 1889.]

a? that the lengthof the arc of the hyperbola xy and x=c is equal to the arc of the between the limits x=b the limits aV2 between curve r=b, r=c. ""2(a4+r4) 13. Show

=

=

[OXFORD, 1888.] 14. Show

that

in the

parabola

"=1+0080,-^ =:-__^_and T

hence

show

that the

extremity of

arc

'

between intercepted

the latus rectum

d"Y

sin^w*

and the_yertex is a{\/2-flog(l +\/2)}.

the

[I.C. S., 1882.]

ETC.

RECTIFICATION, 121.

Length

of the

Arc

of

an

143

E volute.

It has been shown (Diff.Gale, for Beg.,Art. that the difference between the radii of curvature

157) at

Fig. 12.

pointsof a arc corresponding

two

curve

is

equal to

lengthof

the

the

of the evolute ;

if ah be the arc of the evolute of i.e., of the original then (Fig.12) curve, .e.

(atA)

the

"

/"

portionAH (at H),

INTEGRAL

144

and

if the

CALCULUS.

volute be

regardedas a rigidcurve, and a from it,beingkept tight, then the stringbe unwound pointsof the unwinding stringdescribe a system of of which is the original AH. curves one curve parallel e

Find

Ex.

the

lengthof

the evolute

of the

Let a, a',/3, be the centres of curvature /3' the extremities of the axes, viz., A, A', B, B' of the evolute arc a/3 correspondsto the arc and we have (Fig.13) arc

[forrad. Thus

of

the

curv.

of

lengthof

a/2 /o(at 5)-p(at A) =

ellipse =

^.

the entire

=

~-

ellipse. correspondingto respectively.The of the curve,

AB

"

Ex. 3, p. 153,Diff. Beg."]. Calc.for

perimeter of

the evolute

EXAMPLE. in the above manner for the parabolay2 kax that the within the parabola lengthof the part of the evolute intercepted Show

=

is4a(3\/3-l). 122. Intrinsic The

Equation.

relation between

given curve,

measured

s,

the

from

lengthof the arc of a a given fixed point on

Fig. 14.

the curve, and the extremities of the of the

curve.

anglebetween arc

tangentsat the Intrinsic Equation

the

is called the

ETC.

RECTIFICATION, 123. To

the Intrinsic

obtain

145

Equation

from

the

Cartesian. be given as y=f(x). of the curve Let the equation and Supposethe #-axis to be a tangent at the origin, from the origin. the lengthof the arc to be measured

-^=/("),

(1)

s=\ *Jl+ [f'(x)~]2dx

(2)

Then

tan

also

from (2),and x If s be determined by integration eliminated between this result and equation (1),the requiredrelation between s and ^ will be obtained. .

Ex.

1.

be i/r

If

tangent

and

Intrinsic

the at the

equationof

the initial tangent at A and the have the radius of the circle, we

angle between point P,

and

circle.

a

a

therefore

Ex.

s"a^r.

the case equationis s

2.

=

For

of the

In

c

tan

tan^

catenary y

+

^ dx

and

/-,

as

T

"

dx P. I. c,

=

=

ccosh-, c

^. =

c

\/ 1 "

-r

"

i

""x

smh2-

=

c

K

the

trinsic in-

therefore

and

CALCUL

INTEGRAL

146

s

=

c

sinh

US.

-,

c

of the constant whence together,

being chosen integration s

124.

To

obtain

=

c

tan

so

that

x

and

s

vanish

\js,

the Intrinsic

Equation

from

the

Polar.

Fig. 16. /

",

to the the initial line parallel pointfrom which the arc is measured.

Take

usual notation

we

tangent at Then

the

with the

have

=/(0),the equationto 0+0, ^ T

the curve,

=

.......................................

.......

(1) (2)

from (4),and 0, "f" by integration of equations(2) and eliminated (3),the by means requiredrelation between s and \fswill be found. If

Ex.

Here

s

be

Find

found

equationof the r=a(l -cos 0).

the intrinsic

i/r

and a

sin

0

2

cardioide

CALCULUS,

INTEGRAL

148

the

125. When

By

dy

tan

have

we

of

means

of the Curve

Equation

^

=

-^ ax

is

given

d)'(t)

^_-^

=

.......

as

,-

.....

N

(1)

........

j (t)

equation(2)s

be

may

of t. the result and If then, between shall obtain the eliminated, we

found

by

tegrati in-

in terms

and

between

s

Ex.

In the

^.

cycloid y"a(\

we

have

tan

Also

whence Hence

126.

5=4a

5

=

equation(1) t be required relation

sin 4ot sin

t\

-cos

*

*mt

^

=

^ dt

=

tan

= ,

1+costf

2

+ cos02+ ax/(l

=

2a

cos

-,

2

if s he measured

-

sin2*

from

the

originwhere

Z=0.

is the equationrequired. T/T

Intrinsic

Equation

of the Evolute.

be the equationof the given curve. s=f(\f/) Let s' be the lengthof the arc of the evolute measured fixed pointA to any other pointQ. Let from some 0 and P be the pointson the original sponding correcurve to the pointsA, Q on the evolute; p0, p the P: at 0 and radii of curvature \j/the angle the tangent QP makes with OA produced,and ^ the anglethe tangentPT makes with the tangent at 0. Let

ETC.

RECTIFICATION, Then

149

and "*//-^r, =

ds

or

O

*

T

Fig. 18.

Equation of an Involute. if the curve the same With AQ be givenby figure, have the equation we s'=f(\}/), 127. Intrinsic

and whence Ex.

=

intrinsic

The

\Is "

\//,

\ equation

the

of

catenary

is s=ctsm\Ir

(Art. 123). Hence

and

radius

p0= =

' .

.

The

the evolute

c

p

its evolute

=

=

y

c

s

=

so

c

=

s

c

=

T/r=0

tan2^.

is

+ A^r + logsec T/T

that 5=0

measured s

and sec2i/r

I(ctan "fy + A)d^r

=

=

if s be

-

is

at the vertex

of curvature

1 ),or c(sec2-v//intrinsic equation of an involute

is

s

and

equationof

the intrinsic

c

T log(sec

when

constant

^=0,

; we

have

,

INTEGRAL

150

128.

Length of Arc

If p be the

CALCULUS.

of Pedal

Curve.

from perpendicular

originupon the tangentto any curve, and ^ the angle it makes with the initial line, we regardp, % as the current may polarcoordinates of a pointon the pedalcurve. culated Hence the lengthof the pedal curve be calmay by the formula

Ex. of the

Apply the above method to find the lengthof pedalof a circle with regard to a point on the a cardioide). (i.e.

Fig.

Here, if 2a

be the

arc

of

=

pedal

OP

=

the

-

J

/2a

cos

*dx

cos

-a

2

2

*

figure

2ctcos2*.

/2 A/a2cos4+ a2 sin2-

=

=

from

have

* cos

any arc circumference

19.

diameter, we p

Hence

the

=

4a sin

2 + C.

j

The

limits for the upper half of the Hence the whole perimeterof the

curve

1 2[4asin-Jo L-

2

are

pedal =8a.

x

=

0 and

X

=

TT.

ETC.

RECTIFICATION,

151

EXAMPLES. 1-. Find

the

length of

2. Find

the

lengthof the

of the

arc

any

fu\a x)=aP.

curve

"

[a,1888.]

completecycloidgivenby

y the

s

=

a

^

tan

equation of

^r+

sec

the

parabolais

+ sec ^). log(tani/r

a

Interpretthe expressions

the wherein closed given

Jw

0. 1

curve

that the intrinsic

4. Show

5.

cos

"a

for which the lengthof the arc measured the originvaries as the square root of the ordinate.

3. Find

from

a

=

6. The

line

taken

are integrals

rou

id the

a

[ST.JOHN'S, 1890.]

curve.

major

perimeterof

of

axis

is 1/10. ^/^eccentricity

is ellipse

an

1 foot

its circumference

Prove

nearly.

in

length,and

to be 3*1337

its feet

[TRINITY,1883.]

7. Show cardioide

that r

4r=3asec

0 remote

8. Find

length of the arc of that part of the cos 0), which lies on the side of the line from the pole,is equal to 4a. [OXFORD, 1888.]

the

a(l +

=

the

lengthof

of the cissoid

arc

an

r_asin26" cos

9. Find

the

10. Show

lengthof

show

a

that

of the

arc

certain

=

curve

equation of 4a(sec3Vr1).

that the intrinsic

3a3/2=2^ is 9s 11. In

any

ff

-

curve

5=ee\/2+

a

the semicubical

bola para-

Show

12.

is

CALCULUS.

INTEGRAL

152

the

that

given by

s

Show

13.

is 14.

s

Show

between

curve

y

of the

arc

+

that

the

the

curve

C.

Trace

tion equa-

a

of the

length

arc

of the

y=logcoth-

curve

points (xl9yj),(#2,3/2)is log s!n x^. sinn

15.

intrinsic

the

alogsec-

=

gd~l\ff.

a

=

an

=/("9)+/"(#)

in the

that

of

length

the

y2

curve

(a

=

g"

"

#)2,and

X-^

find

the

length

of that

od/

which

part of the evolute

the

corresponds to

loop.

[ST. JOHN'S, Find

the

of

of

and

1891.]

equiangular spiral pole. Show that the arcs of an from equiangular spiral measured the pole to the different with another points of its intersection pole but a different angle equiangular spiralhaving the same will form in series a [TRINITY, 1884.] geometrical progression. 16.

(p

17. has

measured

rsma)

=

length

1881

that

Show

from

the

for its intrinsic

arc

an

an

the

whose

curve

equation

s

is

pedal equation

p2=r2

a?

"

a"-.

=

Zi

18.

is

Show

equal

to the 19. curve

to

that that

the of

that

minimum

the

rm=amsinmO

length

ellipsewhose

an

and

maximum Prove

whole

of

semi-axes

radii of

length

the

the

limagon r=acos are equal

vectores

nth

of the

pedal

of

a

in

length

limacon.

loop of

the

is mn-m+1

,-m

(smmO)

a(mn+I)

m

dO.

^

1883

j

o

20.

Show

that

the

length

of

a

loop

of the

curve

[ST. JOHN'S, 1881.]

X.

CHAPTER

ETC.

QUADRATURE, 129.

Areas.

Cartesians.

bounded process of findingthe area is termed quadrature. portionof a curve The

It has bounded ordinates

by [x

considered

a

=

and

x

the limit of the

as

rectangles;and

two

area

"f"(x)], pair of any

=

the

sum

the

any

axis

of

an

that

of x9 may be infinite number

the

expression

is

area

the

[y

b] and

=

1 ydx

In

line

curved

any

of inscribed for the

in Art. 2 that

already shown

been

by

same

way

the

given abscissae [y

0 (x)dx.

or

=

bounded

area

c, y

d]

=

and

by any curve, the y-axisis

fxdy. by two Again,if the area desired be bounded given curves [y "p(%)and 2/ \^(^)]and two given ordinates \x a and x 6],it will be clear by similar reasoningthat this area may be also considered as the limit of the sum of a series of rectangles constructed 130.

=

=

"

=

INTEGRAL

154

indicated

as

in the

expressionfor the

figure. The be accordingly

will

area

CALCULUS.

Li% PQ

dx

fj"(0) \fs(x)]dx.

or

-

J

x=a

Fig. 20. Ex.

Find

1.

ordinates

the x"d

x=c,

Here

by the ellipse"- + 2

bounded

area

and

^2 =

-

1, the

b2

the

f ^Sr

area=

J

a

2a

For the

we quadrant of the ellipse above expressionbecomes

"

?

a2 .

givingirab Ex.

Find

2.

y2

area

the

area

x2 and

and

between

the

4

of the whole above

c=0

ellipse.

the #-axis

included

y2 ax. parabolatouch at the originand from #=0 are (a,a). So the limits of integration The area sought is therefore

curves

The

at

for the

and

.

2

2a

=a

^"

or

.

put d

must

a

=

%ax

"

circle and

=

the

fa

?

-

x2

~

cut to

x"a.

again

INTEGRAL

156

portionbetween

the

For

limits are For the

CALCULUS.

a

"

to

loopwe

the

0, and double

and

curve

the

asymptote the

before.

as

have

therefore

a+x

for the

portionbetween

the

and

curve

x\l /O

the

In

asymptote,

/v. _

dx.

" a+x

Fig.22. To

integrateIxJa~xdx, put J

*

cos

x=a

Then

a ~p x

and

0

dx"

"a

ftfJEfcfcl\ Va+x -

J

J

r =

and

sin 0 c?$.

area

of

a2/

--}" I)

1-cos2^

ETC.

QUADRATURE,

Again,

rxJ?E*dx=

sign before

of the

the

cos

"^gain fl^1 l-cos2#

negativesignis

radical

in

Bd9

"

J

"a+#

[The meaning +

a

-

J_

157

this

y=#/v/^" _

*

:

"

In

choosingthe

are

tracing the

we

a+x

the curve below the #-axis on the left of the origin and above the axis on the rightof the origin. Hence y being be is it referred to between limits the expected to, negative that we should obtain a negativevalue for the expression

portionof

Thus

the whole

area

requiredis

in this example that the greatest also be observed assumed that infinite one. In Art. 2 it was ordinate is an for the the result area finite. Is then was every ordinate ? t rue and the the bounded curve asymptote rigorously by limits let us integratebetween To examine this more closely small positive e is some a + e and quantity,so as 0, where have to exclude the infinite ordinate at the point x" "a, we

[It must

"

as

before

A/fEfdfc. [""c "

J

*a+x

where so

that 8 is

a

small angle. positive

This

4

which

close approaches indefinitely

when

8 is made

to diminish

without

is integral

2

to the former

limit.]

4

result

INTEGRAL

158

CALCULUS.

EXAMPLES. 1. Obtain

the

bounded

area

by

a

parabola and

its latus

rectum. 2. Obtain

bounded the areas by the curve, ordinates in the following cases : specified

the

and ^7-axis,

the

"

(a) #=ccosh-,

to x=h. x=

a

x=a

the

3. Obtain 4. Find

the

area

bounded

of X2la2+y2/b2l is divided areas

=

5. Find

the whole

area

the

to x=b. to x"b.

by the curves y2=4ax, #2=4ay. portionsinto which the ellipse

the line y=c. included between

by

the

curve

X2y2=a2(y2x2) "

and

its

asymptotes.

6. Find

the

between

area

the

curve

y2(a+x)=(a

"

xf

and

its

asymptote. 7. Find

131.

the

area

of the

Sectorial

Areas.

the

curve

y*x+ (x + af(x + 2a)

=

0.

Folars.

to be found

is bounded by a curve r=f(6) and two radii vectores drawn from the origin divide the area into elementary in givendirections, we small angle89, as shown in the sectors with the same figure.Let the area to be found be bounded by the arc When

the

loopof

area

PQ and the radii vectores OP, OQ. Draw radii vectores OP19OP2, OPn-i at equal angularintervals. Then by drawingwith centre 0 the successive circular arcs that the seen PN, P1NV P2^2,etc.,it may be at once of the circular sectors OPN, OP^N^ limit of the sum OP2N%, etc.,is the area required. For the remaining elements PNPV P^N^P^ P2^2P3,etc.,may be made to rotate about 0 so as to occupy new positionson the ...

ETC.

QUADRATURE,

159

greatestsector say OPn-iQ as indicated in the figure. Their sum is plainlyless than this sector ; and in the limit when the angle of the sector is indefinitely diminished

its

area

the radius

also diminishes

vector

OQ

without

remains

limit provided

finite.

O

Fig. 23.

The

of

area

a

circular sector is

of angleof sector. X circular meas. J(radius)2 the summation Thus the area required l?L"Zr2S(), being conducted for such values of 9 as lie between xOP and 0 6 xOPn-i, i.e., xOQ in the limit,Ox being =

=

=

the initial line. In and

the

xOQ

=

notation

of the

/3,this

will be

calculus integral expressedas

if xOP

=

a,

or

Ex. and

1.

Obtain

the

the initial line. the radius vector

Here

0=0

area

to

0"

-.

Hence

of the semicircle

sweeps

the

area

over

the

bounded

by

r

angularinterval

is

2

=

i.e., ^radius)*.

acos

from

0

CALCULUS.

INTEGRAL

160 Ex. 2.

Obtain

the

of

area

loopof

a

the

curve

sin 3ft

r"a

This curve will be found to consist of three equal loopsas indicated in the figure(Fig.24). The proper limits for making the integration extend over the first loop are 0=0 and 6 of 0 for which r vanishes.

of

.-. area

for these

=

-,

are

successive values

two

^

loop 1 fWn2

30 dO

=

=

f\l

60)d9

-cos

~

3~~ 12'

4

2

The

total

of the three

area

loopsis

therefore!^.

Fig. 24. EXAMPLES. the

Find 1. r2

=

bounded

areas

loopof 4. One loopof loopof r=asin2ft by the portionof r=ae^coiabounded 0=/3 + y (y being less than 2?r).

5. The

9=13 and 6.

Any

sector

of

7.

Any

sector

of

8.

Any

sector

of

cardioide

9. The

s

originand show

3. One

"2cos20+ 62sin2ft

2. One

10. If

by

that

be the

$

=

r=

a

sin 4ft

r

a

shift ft

=

radii vectores

7^0=^ ((9=a to 6*=^). r0""a (0=a to ^=)8). r(9

(9=a to a(l cos 0). a

=

r

=

0

=

fi).

"

lengthof =

between

r="tanh-

curve

the

2

the

27r,and A A

the

area

between

a(s air). "

the

same

points,

[OXFOKD,

1888.

]

ETC.

QUADRATURE, 132. Area

a

on

Closed

a

Curve.

(x,y) be the Cartesian coordinates of any point closed curve ; (x+ Sx,y + Sy)those of an adjacent

Let P

of

161

pointQ.

Let

(r,9),(r+

Sr , 6 +

$0)be the corresponding

shall suppose that in Also we polar coordinates. from P to Q along the along the curve travelling infinitesimal arc PQ the direction of rotation of the OP that the is counter-clockwise radius vector (i.e.

Fig.25.

hand to a person the left Then the element direction). is

area

on

AOPQ

ir2(S$

=

Hence

another is

curve

in this travelling

$(xSy ySx).

=

"

for expression

the

area

of

closed

a

f

Wxdy-ydx), beingsuch completelyround the the limits

133.

may

If

write

we

put

the above

y

that the

point(x,y) travels

once

curve.

=

i

so

that

^M^ =

(fo)we

where as ^\xzdv, expression

is

x

the limits of integrati that the current point(x,y) travels so chosen As v is really once completelyround the curve. 6 is a rightanglecare tan 6 and becomes infinitewhen be taken not to integrate must throughthe value oo to be

expressedin

terms

of

v

and

.

E. i. c.

L

INTEGRAL

162 Find

Ex.

by

CALCULUS.

this method

the

of the

area

ellipse

#2/a2+.y2/"2=l. Putting y

=

vx,

have

we

and

*

=

f"^L= f-

JV between

Now,

and

properlychosen

limits. in the first quadrant v varies from area

of

quadrant =?"

area

of

ellipse

therefore

134. If the

originlie

point P

current

elements such

L2

*

as

"

0 to

oo

Hence .

-,

=irab.

without

travels round

the

curve,

as

the

obtain

triangular of space such as OP1QV includingportions OP2Q2 shown in the figurewhich lie outside we

Fig.26.

the

curve.

removed travels element and

S6

These portionsare however ultimately from the whole integralwhen the point P the element over P2Q2, for the triangular

as OP2Q2 is reckoned negatively is negative.

135. If however

^ I(xdy

"

the

ydx), taken

longerrepresentsthe

curve

cross

round sum

the expression itself,

the whole

of the

9 is decreasing

areas

perimeter,no of the several

CALCULUS.

INTEGRAL

164

the curve is when speciallyadapted to the cases defined by other systems of coordinates. Ss of a plane curve, and OF If PQ be an element the chord the pole on from the perpendicular PQ,

Fig. 28.

|OF.PQ,and any sectorial area summation the along the whole being conducted of the IntegralCalculus In the notation bounding arc. AOPQ

=

=

this is

be at

[Thismay

deduced

once

from

| rW,

(V2d0ir^ds sin 0 \r =

=

(where "f"is

angle between

the

radius vector)

137.

Tangential-PolarForm. .

.

ds

Again, we

have

since

area

P

=

=

=

5^

d*p

P +

\ \pds

=

J

the

thus

:"

ds

tangent

and

the

ETC.

QUADRATURE, a

formula

suitable for

when

use

165

Tangential-Polar

the

equationis given. 138. Closed When

the

Curve. is closed this

curve

admits expression

of

simplification.

some

For and term

perimeterthe

the whole

in

round integrating disappears.Hence

when

the

curve

is dosed

first we

have area

^

=

the equationof Ex. C ale.forBeginners, By Ex. 23, p. 113, Diff. the one-cusped be the (i.e., cardioide) expressedas epicycloid may p

=

Fig. 29. Hence

its whole

limits

i/r 0 =

a2cos2^\d^ taken area=-^/f 9a2sin2;' "

and

^="

Putting-^ 3$,this =

and

doubled.

becomes IT

=

3a2

f (9sin2^ ^o

-

co**0)dO =

67ra2.

tween be-

INTEGRAL

166

Pedal

139.

Again,

CALCULUS.

Equation.

for

pedal equations,

have

we

A

ip

=

Ex.

ds

In the

Hence

dr

i p

=

sectorial

any

f

of curvature case

J

we

and

area

Ex. to the

1.

t

=

The

circle is

area

i

-

a,

rcosa

between evolute.

take

as

a

curve,

element

our

by

two

the infinitesimal

arc

infinitesimals

first order

sin

=

y2si

elementary trianglecontained of curvature

0 dr

sec

/

included and the

140. Area

this

}p

area

/"2

In

=

equiangularspiralp=r

=

To

their

given by

curves

.e.

p, between

(Fig.31)

a

this

p\

is

radii

two

of

the

area

contiguousradii ds of the

curve.

and |/o2"S\^,

the

or

its involute,and circle,

a

tangent

ETC.

QUADRATURE,

167

the tractrix and its asymptote is between Ex. 2. The area found in a similar manner. such that the portionof its tangent The tractrix is a curve and the ^7-axis is of constant between the point of contact

length c.

Fig.31.

Taking two adjacenttangents and elemental triangle (Fig.32)

o

the axis of

T

x

as

forming an

r

Fig. 32.

EXAMPLES. 1. Find

the

area

of the

[Limits\jf 0 =

2. Obtain

the

same

two-cuspedepicycloid

"^=7rfor

to

result

by

means

7.2 ^2 + =

[Limitsr=a

to

r

=

one

quadrant.]

of its

pedalequation

1^2.

2a for

one

quadrant.]

CALCULUS.

INTEGRAL

168 3. Find

the

the radius

between

area

of curvature

the

at

catenary s the vertex, and

=

c

^,

tan

any

its

other

evolute, of

radius

curvature.

the

4. Find

evolute,and 5. Find

evolute,and

area

any

between

the

epicycloids

=

AsmB^s,

its

equiangularspirals"Ae^y

its

radii of curvature.

two

AREAS of Pedal

141. Area

the

radii of curvature.

two

any

the

between

area

OF

PEDALS.

Curve.

If

_p=/(Vr)be the tangential-polar equation (Diff. Gale, for Beginners,Art. 130) of a given curve, S\fs will be the angle between the perpendiculars two on contiguoustangents,and the area of the pedal may be expressedas

(compare Art. J|p2c^/r

131).

Fig.33. Ex.

Find

the

area

of the

pedal of

a

circle with

regardto

a

(thecardioide). pointon the circumference if OF be the perpendicular Here the tangent at P, and on OA the diameter obvious that OP ( 2a), it is geometrically =

bisects the the

Hence, callingYOA"^, angle AOY. tangential polarequation of the circle

Hence

=

^/

we

have

for

ETC.

QUADRATURE,

169

where the limits are to be taken as 0 and TT, and the result to be doubled so as to include the lower portionof the pedal. Thus

*cos*fe^ 4a2. ^l=4aaf 2 J =

o

2

f 4222

J o

Fig. 34.

142. Locus Let

0 be

a

o

Origins of

Pedals

of

given

Area.

fixed

ordinates point. Let pt \jsbe the polarcothe foot of a perpendicular OF upon any givencurve.

a

of

tangentto

of

0

Fig.35.

Let P

be any from

other P

upon

fixed the

point,J"F1(=^1)the perpendicul tangent. Then the areas

INTEGRAL

170 of the

CALCULUS.

0 and

pedalswith

P

as originsare respectively

and j[ftL2"% ijV2cfyr taken

between A

and Al coordinates of P areas

Cartesian Pi and p is 2 Al

Call limits. respectivelyLet r, 6 be the with

regard

equivalents.Then T cos($" -t/r) P =p ~~

"

a

function

known

\(p \p^d\fr

=

definite

certain

=

"

x

of

cos

x

"

cos

ifs "

x9

y sin

y

their

i/r,

\fs Hence

i/r "

"

y sin

\l/fd\^

"

1cos + x* Icos2i/r + 2a32/ c?i/r

si \[s

1si 2/-2

+

2

0, and

polar

Vp^d"^2x \pcos ^ d\/r 2^/|psin \fsd\fs

=

Now

to

these

I 2 Ipsin \/r d\[s, d\fs, \/r

Jp

cos

such limits that the whole pedal is described Call them will be definite constants.

between

-20, and

2Al

=

If then P move its locus must

"2A +

Hence

a

known

2gx + 2fy+

2hxy + by2. that Al is constant,

ax2 +

in such a manner be a conic section.

143. Character It is

2A, 6,

a,

obtain

thus

we

-2/,

of Conic, result in

it will be obvious

that inequalities

that if p, q, r,

...

stand ,

for

INTEGRAL

172

will

removed.

thereby be

if II be the

vanish, and

for any

have

we

2A1 generalcase.

in the

"2 is

Thus

of the

area

point such

211 + ax2 +

=

The

of this conic is

area

"

*/ab

,

""

"

s

"

I?

\

(area or conic).

, --

Thus

h2

"

TT

-d.1 IH =

the

pole

2hxy + by*

(Smith'sConic Sections,Art. 171). A

pedal whose

other

*

For

a

both and \psiu\^d\[^ integrals\pcos\fsd\fs

that the

is "2

CALCULUS,

^7T

of any

case particular

closed oval the equation

of the conic becomes

whence

J.1

where

is the radius of the circle

r

values of

constant

on

which

i.e.the distance of P

Av

P

from

146. Position

of the

Point

"2 for Centric

In

which

has

centre

oval

any

plainlyat

a

that centre, for when

the

the centre

lies for "2.

Oval.

point "2 is taken

is as

and \psui\fsd\ origin,the integrals\pcos\fsd\fs both

vanish

complete oval cancelling), 147-

Ex.

Here and Hence

1.

Find

the

the

area

point within (a limagon).

regard to centre

is performedfor the integration (oppositeelements of the integration

when

the

any

A n

=

=

of the pedal of a circle with circle at a distance c from the

n+^, 7ra2.

Ai=ica*+"

.

ETC.

QUADRATURE,

of the pedal of an to any point at distance c from the centre. of the pedalwith In this case II is the area Ex.

Find

2.

the

/* Vcos2"9 + b%m*0)dO

2

-

area

Ex.

c

with regard ellipse

regardto

the centre

"2)|.

(a2+

^1=|(

Hence

taken

=

173

The

3.

with

from

area

pedalof the cardioide r=a(l "cos 0) internal pointon the axis at a distance

of the

respectto

an

the poleis

|(5"s-2"c+2c'). [MATH.TBIpos"187a] pole,P the given internal point; p OF2 and PTl on any tangent perpendiculars

Let 0 the two

and P pl "p

be

respectively ; ""the angle Y$P "

c cos

and pl from 0

the

"",and

^Al=2A

"

and

2clp

cos

OP"c

; then

+ / ""cfc"

Fig. 36. Now

in order

that p may

between integrate the cardioide

limits

sweep

"" =

0 and

the whole

out

"" =

-^and

pedalwe

double.

Now

must

in

(Fig.36) p=

OQ

sin

Y2QO

=

OQ

sin^xOQ. [Dif.Calc.,p. 190.]

CALCULUS.

INTEGRAL

174

For

r2"0

itf0"| =

=

|-{*-("-W-|

Hence

|-*=f,

or

J-J-J,

-

/3

so

"

-

-.

23

A*-

,

Hence

/p

cos

"j" d"j" 2 =

,

2a cos3 2

/

cos

d(f" "/"

cos

3,so?2

'

=

fl

4a

3

x

/ cos% o

=

rf [4cos%

/

12#

3

-

cos")"iz

42

6422

Also

2

^^^ fc2cos2d"cta=3.2c2i J

222 Sir

Finally

24

=

2

Tcos^"fe, 4a*"*^*J"

J

=

6

1

3

642 mi

A Al

When

f2 is taken 2A

=

l

Hence

as

it is positive,

'

T"

8~

of Minimum

Area.

it appears origin,

that

Pedal as

211 +

the term

?rac

--

Origin for

^

2

_?ra

148.

24a2

J(05

cos

^

+ y sin

\Jsfd\ls.

is necessarily \(xcos\fs + ysm\}s)"2d\fs

clear that

Al

can

never

be

less than

II.

ETC.

QUADRATURE, "2 is therefore

pedal curve 149.

has

Pedal

in Art. 138

minimum

a

of

which

corresponding

the

area.

Evolute

an

for the

formula

The

originfor

the

175

of

Closed

a

of any

area

Oval.

proved

closed oval

is of

area

jp2*/' Jjft)

Hence

J

=

oval +

plainlyexpresses that the area of any pedal of the oval itself is equal to the area of an oval curve togetherwith the area of the pedalof the evolute (for which

-ry

is the

radius

also admits

This

vector

of

area

of

above

article shows

pedalof

evolute

the

evolute).

proof. elementarygeometrical

of the Find the area Ex. with regardto the centre. The

pedalof

of the

=

-

pedalof

the

evolute

of

an

ellipse

that area

of

pedal of ellipse area -

-(a2+ b2) -

irab

=

?(a b)2. -

of

ellipse

INTEGRAL

176 150. Area of

pair

bounded

by

a

Curve, its Pedal, and

a

Tangents.

P, Q be

Let

CALCULUS.

two

contiguouspoints on a given 7, F' the corresponding pointsof the pedal of curve, since (with the usual notation) Then any origin0.

PF=-vjrthe elementarytrianglebounded contiguoustangentsPY, QY'

and

the chord

by YY'

two

is to

the firstorder of infinitesimals

Fig.38.

Hence

the

area

curves

and

a

may

be

and

is the of the

area

151. Let

of any portionbounded by the two curve pair of tangentsto the original

expressedas

same

as

the

pedal of the

portion of the corresponding evolute.

Corresponding Points f(x,2/) =

0 be

any

and

closed

Areas.

curve.

Its

area

(A^)

QUADRATURE, is

177

\ydxtaken line-integral

the

expressedby

ETC.

round

the complete contour. If the coordinates connected relations the curve

by

of the current point (x,y) be with those of a second point(" rj) by the mg, y nrj, this second pointwill trace out is expressed /(w" nrj) 0 whose area (J.9)

x

=

=

=

Irjdg taken line-integral

the

And

its contour.

have

we

\ydx

l=

whence

round

=

\nrjm

=

it appears that the area of any 0 is mn times that of f(inx, ny)

f(xty)

=

152. Ex.

1.

Apply

this method

**

,"*

+

~

=, a

b

r

to find the

=

curve

0.

of the

area

ellipse

1

r

the

point", T\ traces corresponding

and

area

of

closed

out the circle

ellipse

x area

=~

of circle

r

Ex.

Find

2.

the

area

Let

in

curve

^

ny

mx

then the

or

of the =

the central E. i, c.

pedal of

r2

=

n^f)2 =

a

V

+

ij,

^- cos2 0 + m2

an

=

+

is

correspondingcurve

polars

(mV

sin20, ri2 "--,

symmetrical about ellipse, M

both

axes.

Wif-

INTEGRAL

178 Hence

the

CALCULUS.

of the first curve

area

=

x

"

of second

area

mn

EXAMPLES, 1. Find

the

of the

area

loopof

the

curve

ay"L=x\a-x). 2. Find

the whole

of the

area

[I.C. S.,1882.]

curve

a?y2 a2x2-x*. y\ and a2#2=;?/3(2ct

[I.C. S., 1881.]

=

3. Trace

the

curve

that of the circle whose

equalto

that its

prove

"

radius is

the curve cfiy*xb(Za x\ and prove radius is a as 5 to 4. to that of the circle whose 5. Find

=

the whole

and

that its

"

of the

area

is

a.

[I.C. S., 1887 4. Trace

area

1890.] area

is

curve

[CLARE, etc.,1892.] 6.

By

of the

show

means

integral/y

of the

formed triangle

that

by

they enclose

the

the

taken

dx

round

the

contour

lines intersecting

area

[Sir.PKIZE, 1876.] 7. Find

the

area

between

and

y2

=

a

8. If

show

ty be

that the

the area

of y

an

oval

cos

ds "fy

integration being taken

makes

with

the axis of #,

is

curve

or

its asymptote.

x

angle the tangent

/r the

"

q:

all round

/x the

sin

^rds,

perimeter.

INTEGRAL

180

CALCULUS.

line to a point P on the curve if A be the ; and by the curve, the initial line,and the radius vector 9,42 Find

21.

the

P,

to

then

2rf.

=

of the closed

area

bounded

area

3a sin

portionof 0

the Folium

9

cos

_

~sin^6"TcoW ratio does

what

In

the

line x+y

[I.C. S., 1881] divide

Za

=

the

loop? 22.

of the

area

[OXFORD, 1889.] Find

the

of the

area

given radii vectores

and

r=aOebe

curve

between

enclosed

branches

successive

two

of the

two

curve.

[TRINITY,1881.] 23.

Find

the

24. Show

^" -,

and

that

the

of

area

the total

state

loop of

of the

area

the

loop of

a

in the

area

curve

r

a0cos 0 between

=

the

curve

n

odd, n

cases

r

=

acosn0

is

even.

4?i

25.

Find

the

of

area

loop of

a

the

curve

r

=

a

cos

3$ + b sin 3$.

[I.C. S., 1890.] that the 26. Show the curve r=acos5$

contained

area

is

between

the circle

equal to three-fourths

of the

circle. that the

of the

area

-

equal to

curve

30.

its

the =

Find r

=

9+

cos

a2sin2#) aV =

[I.C. S., 1879.]

irac.

Find

equation r 29.

area

curve

sin 0

2ac r2(2c2cos2"9

28.

of the

[OXFORD, 1888.]

27. Prove

is

and

r=a

whole

0+

acos

the

of

area

the

b,assuming included

area

b

"

a.

between

the

two

a(2cos 0 + ^3).

Find

the

loops of [OXFORD,

between

area

the

the

representedby

curve

1889.

r=a(sec $+cos 0)

curve

the

]

and

asymptote.

31.

Prove

lemniscate

that r2

=

the

a2cos2$

area

of

with

respect to the

loop

one

of

the

pedal

of

the

poleis a2. [OXFORD, 1885.]

32.

Find

the

area

of the

loop of

the

curve

(x'\-y)(x^+y2)^axy.

[OXFORD,

=

33.

Prove

that the

area

of the

loop of

the

curve

1890.]

QUADRATURE, Find

34.

and

its

35.

the whole

that

between

vector

r

36.

between

the

of the

Show

the

the

the

centre, is

a2+b2-r2 ellipse(*L^ p2

the

semi-major axis,and

tan"1^/^^-, a,

"

0 and

=

its

between

being

the

the

a2log(sec ""+

-

curve

5

=

atan^,

V*"""""ig

at

tangent

^ + a2 tan "/"

-a2 tan

b

radius

a

[CLARE, etc.,1882.]

in eluded

area

in-

=

ellipse.

^

at

of

area

curve,

that the

tangent

curve

[OXFORD, 1888.]

the

from

semi-axes

its

contained

area

181

asymptotes.

Show

eluded

ETC.

tan

c"). [TRINITY, 1892.]

37. Show p =^isin 38.

that

Sty and

Show

whose

the

the epicycloid space between taken from cusp to cusp is ^irA2B.

its pedalcurve

that

areas

the

of

area

the

a2(fTT

are

a(^\/3 + cos^#) has three loops 2\/3), a2(f f\/3), spectively. a\ -far fV") re-

curve

r

+

=

TT

-

-

[COLLEGES, 1892.] 39. Find

the

of

area

loop of the curve x*+y* Za2xy.

a

[OXFORD, 1888.]

=

40.

Find

the

of the

area

the

pedal of

curve

d*)l, =*("**the

originbeing taken

41.

the

Find

curve

42.

the

3%2

Find

at

x

=

*Ja2 62,y "

included

area

between

and a2(#2+;?/2)

=

the whole

its

of the

area

=

0. one

[OXFORD, 1888.] of the

asymptotes.

Find

the

curve

tf+yi a\x*+y*). of a loop of the curve (mV + n,y)* aV b2y2-

area

=

44. areas

Trace

the

shape

of the

of

[a,1887.]

[a,1887.]

=

43.

branches

-

followingcurves,

[ST.JOHN'S, 1887. ] and

find

their

: "

(i.)(^+^2)3 =aay*.

(^2+ 2/)3 a^?/4. (ii.) =

[BELL, etc., SCHOLARSHIPS, 45.

Prove

that the 3?

V2 '

of

area '

1 / X2

V2\2

"

7TC2/

1887.]

INTEGRAL

182

46.

Prove

that

the

CALCULUS.

in

area

the

positive

(av+w^w

of

quadrant

the

curve

^(5+5).

is

[a;18900 47.

Prove

that

the

of

the

f")

is

area

curve

-3""

(V2

+

-

tan-1

a2)

-

.

[ST. 48.

Prove

that

the

of

area

the

JOHN'S,

1883.]

curve

9,aV h

62 where 49.

is

c

is

Prove

less

both

than

that

the

area

a

and of

5, the

is

7r(ab

"

Prove

[OXFORD,

^4-3o^3

curve

fTrtt2. 50.

c2). +

[MATH. that

the

areas

of

the

loops

two

24^3)

are

(32^

and

(167r-24\/3)a2,

+

of

the

1890.]

a2(2^2+y2)=0 TRIPOS,

1893.]

curve

a2,

[MATH.

TRIPOS,

1875.]

CHAPTER

SUKFACES

XI.

OF

SOLIDS

about

the

VOLUMES

AND

OF

KEVOLUTION.

It

of Revolution

Volumes

153.

revolve

about

ordinates

x

=

Art.

in

shown

was

the

of

axis

x^ and

x

that

5

the

x

is to

x2

=

if the

a"axis.

curve

y=f(x)

portionbetween be obtained by

the the

formula *2

dx.

Tr2 .

154. More

AB,

and

About

any

axis.

generally,if if PN

be

the

any

revolution

be about

perpendiculardrawn

any

line

from

a

CALCULUS.

INTEGRAL

184

point P on the curve upon the line the volume contiguous perpendicular, if 0 be

or

155. Ex.

loopof

the

Here

1.

Find

the volume

formed

volume

=/

J

J

Let the Then and

a

5a2z

-

+

the revolution

3) about

of the

the tf-axis.

+ x

2az2

~

3 _J"

tion of the spindleformed by the revoluabout the line joiningthe vertex to one

the volume

parabolicarc

extremity of

as expressed

this becomes

z

Find

a

o

rf2a3 log Ex. 2. of a

P'N'

I x*a~xdx. 7ry2dx=7r J

o

Puttinga +x=z,

by

f=x2?"^ (Art.130, Ex.

curve

is

and

the line AB

givenpointon

a

AB

the latus rectum.

parabolabe

the axis of revolution

Fig.

40.

y2

4o

=

is y

=

2^7,

P

"fi

REVOLUTION.

OF

VOLUME

185

Also

and

volume

dAN

= .

o

4?r

75

156.

Surfaces

Aain, if S be out

of Revolution. the curved

y the revolution

of any

surface of the solid traced arc

AB

about

the

^c-axis,

Fig. 41.

PN, QM two adjacentordinates,PN being the of the smaller,3s the elementaryarc PQ, SS the area

suppose

elementaryzone

traced

out

by

the revolution

of

PQ

CALCULUS.

INTEGRAL

186

about

the

#-axis, y and

ordinates of P and Q. Now take we may out

would

be if each

it

axiomatic

as

point of

it

the axis, and less than distance QM from the axis.

SS lies between

therefore

in the limit

8s and

^y

This may

as

from

greaterthan

it

distance PN

pointwere

at

a

2w(y + Sy)Ss,and

8

2-7T2/

"

\

as

to

be

values

of

in any

convenient

particular

beingobtained -j-" -^, etc.,

-r-"

the differential calculus.

157. Ex. formed

area

f

be written

example,the

the

the

have

we

or

=

happen

may

that

at the

if each

r/^ -j-

is

were

from

Then

lengths of

the

in its revolution

by PQ

traced

Sy

y +

by

Find

1.

the revolution

Here

surface

the

of the

of

a

belt

y2

curve

=

"ax

of the

paraboloid

about

the #-axis.

about

the

=

V x

dx

dx "/"

y"dx /X"1 dx

Ex. line. Here

The

2.

Find

curve

r

the volume

volume

=

=

=

a(l+

and

/try^dx =

TT

cos

surface TT

6}

revolves

of the

figureformed.

/?'2sin20d(rcos 0)

/a'2(l 0 -f cos2$), -{-cos #)2sin2#a c/(cos

initial

INTEGRAL

188 2. A

Show

and

CALCULUS.

of radius a, revolves round quadrant of a circle, that the surface of the spindlegenerated

that its volume

=

its chord.

-^-(10 3?r). -

3. The

part of the parabolayL "ax cut off by the latus revolves about the tangent at the vertex. Find the surface and the volume of the reel thus generated.

rectum

curved

=

THEOREMS 158. I. When line in its own

OF

any

PAPPUS closed

OR

curve

GULDIN. revolves

about

a

plane,which does not cut the curve, the volume of the ring formed is equal to that of whose height and a cylinderwhose base is the curve is the lengthof the path of the centroid of the area of the curve. Let the #-axis be the axis of rotation. area

with

Divide

the

elements up into infinitesimal rectangular sides parallel such as to the coordinate axes,

(A)

Fig. 43.

each PjPgPgP^,

of

area

SA.

Let the ordinate

PlNl y. infinitesimal angle

Let rotation take placethrough an 89. Then the elementarysolid formed

is

on

=

base

SA

its height to first order infinitesimals is ySO,and therefore to infinitesimals of the third order its volume and

is SA

.

THEOREMS

If the

OF

be

rotation

PAPPUS.

through

obtain by summation SA y If this be integratedover have for the volume curve we .

.

189

finite

any

angle a

we

a.

the

whole

area

of the

of the solid formed

a!i/cL4. Now of

a

for the ordinate the formula of masses number m2, ..., mv

of the centroid with

ordinates

X?7? II

2/i"2/2'""""

-^ then

is y=

seek

we

"

y

the value

of

of the curve, each the ordinate of centroid of the area element 8A is to be multipliedby its ordinate and the sum of all such productsformed, and divided by the

or

of the elements,and

sum

in the

language of

the

have

we

Calculus Integral

A (yd (yd A J

y

=

_

=

i -

.

\dA

A

Thus Therefore But is the

A

volume is the

formed of the area

lengthof the

=

path

A(ciy). revolvingfigureand

This establishes the theorem. COR. If the curve perform have and form a solid ring,we a

=

2-7T and

ay

of its centroid.

volume

complete revolution,

a

=

A(2jry).

closed curve revolves about a 159. II. When any line in its own plane which does not cut the curve, the curved surface of the ring formed is equalto that

INTEGRAL

190

CALCULUS.

and whose of the cylinderwhose base is the curve heightis the lengthof the path of the centroid of the perimeter of the curve. Let the #-axis be the axis of rotation. Divide the perimeters up into infinitesimal elements such as PXP2 each of lengthSs. Let the ordinate PlNl be called y. Let rotation take placethroughan infinitesimal angle S9. Then the elementary formed is ultimately area a with rectangle of the second

sides Ss and ySO,and to infinitesimals order its area is Ss y"9. .

Fig. 44.

If the rotation be through any finite angle a we obtain by summation Ss ya. If this be integrated the whole over perimeterof the curve have for the curved surface of the solid we formed .

an/cfe. If

of the seek the value of the ordinate (rj) centroid of the perimeterof the curve, each element of Ss is to be multiplied by its ordinate,and the sum we

THEOREMS

OF

divided

productsformed, and

all such

the elements,and

US.

PAPP

191

by

the

sum

of

have

we

Lt

languageof

in the

or

Calculus Integral

the

^yds\yds n

\ds

\yd8=8tj,

Thus the surface formed

and

s

s(afj).

=

and perimeterof the revolvingfigure, arj is the length of the path of the centroid of the perimeter. But

is the

s

This establishes the theorem. Con. If the curve perform form

and

solid

a

have

ring,we surface

completerevolution

a =

a

2?r and

s(2 -73-77).

=

and surface of an anchor-ringformed by radius about of circle a line in the plane of a a the circle at distance d from the centre are respectively Ex.

The

volume

the revolution

of

volume surface

Tra2X 2?rc? 27T2a2o?, =

=

2:ra

=

x

Zird

=

4ir2ad.

EXAMPLES. 1. An

major 2. A an

revolves ellipse

axis.

Find

extremity

volume 3. A

which

the volume

revolves

square

about

of the

about

other

the tangent at the of the surface formed.

end

of the

to a diagonalthrough parallel diagonal. Find the surface and a

formed. scalene does not

surface and

trianglerevolves about any line in its plane the cut triangle. Find expressionsfor the

volume

of the solid thus formed.

INTEGRAL

192

Revolution

160. When

we

OAB

area

the

Area.

|r"2o0.Being ultimatelya centroid is f of the way in a completerevolution

from

such

up into OPQ, whose

as

infinitesimals

by

triangularelement, its 0 along its median, and

the centroid

r sin 6) 27r(f

the

revolvingarea

to first order

denoted

about

revolves

sectorial elements

be

may

Sectorial

a

divide

may

infinitesimal area

of

sectorial

any

initial line

CALCULUS.

or

travels

a

distance

the volume

traced

f irr sin 9.

Fig. 45.

Thus

by

Guldin's

by

of this element

the revolution

to first order

traced

by

first theorem

and infinitesimals,

161. If x we

we

=

OAB

sin 9 d9. 7r[r3si

y

=

rsin9,

r3sin 9 S9

have

area

the volume

put

rcos9,

=

therefore

of the whole

the revolution

f

is

r3sin 9 .

7

and

=

r3sin

=

r*cos*9t St

"

=

(9$(tanlf) ~

=

xH

St,

is

EXAMPLES.

and the volume

193

therefore be

may

as expressed

(xHdt. EXAMPLES. and surface of the right the volume 1. Find by integration circular cone formed by the revolution of a right-angled triangle about a side which contains the rightangle. 2. Determine

generatedby

the entire volume of the ellipsoid which is the revolution of an ellipse around its axis major.

[I.C. S.,1887.]

that the volume of the solid generated by the round revolution of an its minor axis,is a mean ellipse portional probetween of the those generatedby the revolution and of the auxiliary circle round the major axis. ellipse 3. Prove

[I.C. S., 1881.] that the surface of the prolate 4. Prove spheroidformed by the revolution of an of its major e about ellipse eccentricity axis is equal to 2

.

of

area

ellipse .

formed Prove also that of all prolate spheroids of surface.

ellipseof

an

the revolution the greatest

by

sphere has

[I.C. S., 1891.]

5. Find

the

given

area, the

of the solid

the volume

loopof

the

y^"x^

curve

producedby

about

the revolution

the axis of

of

x.

[I.C. S., 1892.] 6. Find revolution

the surface and volume of the reel formed of the cycloid round a tangent at the vertex

7. Show that the volume of the cissoid y2(2a "

of the solid formed tion by the revolu^)=x3 about its asymptote is equal

to 2?r2a3. 8. Find

the

curve

the

by

[TRINITY,1886.] the volume of the solid formed by the revolution of (a x)y2 a?x about its asymptote. [I.C. S. 1883. ] -

=

,

9. If the

show

curve

r

=

that the volume greaterthan b. E. i. c.

a

+ b

cos

0 revolve

about

the

initial

line,

4- b2)provided a generated is "7ra(a2

be

[a,1884.] N

CALCULUS.

INTEGRAL

194

the

Find

10.

about

the

and

that

Show

11.

The

if the

of

loop

by

the

the

within

lying

area

the

curve

the

Zay2=x(x

curve

Find

y=a.

revolution r^

=

1890.]

[OxTOKD,

=

of

loop line

straight

formed

parabola r(l+cos $) 2a, volume generated is 187ra3.

line, the

12.

solid

the

the

without

initial

of

0=|.

and

0

=

the

of

radius

prime 6

between

volume

the

volume

"

of

cardioide

revolves

about

the

1892.]

[TRINITY,

1890.]

[OXFORD, Show

13.

of

area

that

the bounded

r=f(0)

of

coordinates the

by

the

the

about a)2 revolves the solid generated.

centroid

0=a,

vectors

the

sectorial

ft,has

for

of

6

=

its

coordinates

f 14.

on

the

Show

the

that

initial

line

at

centroid

a

of

the

distance

cardioide

from

the

-

r

a(l"

=

$)

cos

is

origin.

6 15.

cardioide

the

If

p=rcos(9

"

y\

prove

r

that

=

a(l the

3^7r%2 16.

about volume

cos

#)

volume +

revolve

round

generated

f 7T2"3cos

[ST. JOHN'S,

y.

curve

27r2a3(l+e2). 17.

The

lemniscate

r2

=

a2cos2#

revolves

pole.

Show

that

the

volume

generated

line

1882.]

revolves that

the

[I. C. S., 1892.1 about _

the

the

is

is very 0\ where small, r=a(l -ecos e the initial line. Prove to parallel tangent of the solid thus generated is approximately

The a

"

is -

a

tangent

at

INTEGRAL

196

CALCULUS.

of the

is Sx.Sy, and its mass rectangleRSTU be regarded (to the second order of smallness) may as 0(0,y)Sx Sy. Then the mass of the stripPNMV may be written area

in

or

conformitywith

the

notation

of the

Integral

Calculus

between

the limits y

y =f(x). In performing this integration (withregard to y) x is to be regarded

the

masses

i.e.the

mass

If then all such

we

we

search

are

for the

mass

may

AJKB

area

be summed which the above must the ordinates AJ, BK, and the result may

be written

with the limits of the integration b. from x a to x =

of the

stripsas

lie between be written

which

0 and

findingthe limit of the sum of of all elements in the elementarystripPM, of the strip PM.

constant, for

as

=

=

regard to

x

being

INTEGRATION.

DOUBLE

Thus

mass

of

197

area

A JKB

=

n

or

164. Notation. This is often written

ff "j"(x,y)dxdy, order. the elements dx, dy beingwritten in the reverse There is no uniformlyaccepted convention as to the order to be observed,but as the latter appears to be shall in the the more used notation,we frequently presentvolume adoptit and write

'x,y)dxdy when with

we

are

to be made to consider the firstintegration

regardto

y and the second with

when the firstintegration is with is to say, the right-hand element

regardto

regardto

x.

x,

and

That

indicates the first

integration. If the surface-density of a circular disc bounded by xP+y2 a2 be given to vary as the square of the distance from the y-axis, find the mass of the disc. "JL^ ^ Here we have [juv2 of the element 8x 6y,and its for the .mass is therefore /*#2"#6y, and the whole mass will be mass Ex.

=

// limits for y w411 be ;?/ 0 to y=*Jdl xL for the positive and for x from #=0 The result must then to x=a. quadrant,

The

=

"

INTEGRAL

198 be

multipliedby 4, for the four quadrantsthe the first quadrant.

CALCULUS.

the

distribution

being symmetrical in

of the whole

mass

is four times

that of

Fig.47. and

Putting x=asm0

165.

Other

The

same

dx=acosOdO,

Uses

of Double

theorem

may

be

we

have

Integrals.

for many other few illustrative examples, used

purposes, of which we givea which may the field to indicate to the student serve of investigation But our now scope in open to him. the presentwork does not admit exhaustive treatment of the subjects introduced.

DO

Find

Ex.

UBLE

TION.

INTEGRA

of

the statical moment r2

199

quadrant of

a

the

ellipse

4,2

_+"_ 1,9 a29 62

=

!

'

the surface-density beingsupposeduniform. y-axis, Here each element of area 8x8y is to be multipliedby its surface densitycr (which is by hypothesisconstant in the case and the sum supposed)and by its distance x from the y-axis, the whole of such elementary quantitiesis to be found over will from The be limits of the quadrant. integration y=0 to about

the

7

y

_Va2

=

"

x* for ?/ ; and

from

#=0

to

for

x=a

x.

Thus

we

have

a

moment

/

=

crxdxd'u=^"\Wa2

/

J J

a

"

x2dx

) 0

00

_"rbr _(a?-x2)%-\a_o-ba* aL

Gentroids.

166.

formulae

The

3

Cartesians.

proved

of the centroid of at

Jo

3

in statics for the coordinates

of

number

a

masses

mv

m2, m3,

.

.

.

,

points(xv y^, (x2,y2),etc.,are "

_ ~

apply these to find the coordinates of the centroid of a given area. (See also Arts. 158, 159.) be the surface-density For if at a given point, We

may

o-

then

or

Sx

Sy is the

of the

mass

-

_ "

or,

as

it may

be written

element,and

S(crox 8y)x I("rSxSy)9 when

the limit is taken

I dy \\" arx

dx

J\ardxdy

INTEGRAL

200

CALCULUS,

jja-ydxdy Similarly

~

ff~

\\a-dxdy j J

the limits of

integration beingdetermined will

summation

be

effected for

the

that the

so

whole

in

area

question. Find Art.

the

centroid

of the

quadrant elliptic

of the

Example

in

165.

It

proved

was

moments

there about

/ /"rdxdy=

Also Hence

that the

y

167. Moments When

=

limit

y-axiswas

of the

of the

sum

mentary ele-

?" "

.

quadrant=^^-.

of the

mass

*="

Similarly

the

=-"

3/4

STT

"

of Inertia.

is multiplied by the every element of mass of its distance from a given line,the limit of of of such products is called the Moment

square

the sum Inertia with regardto the line. Such quantities of greatimportancein are Ex.

Dynamics.

Find

the moment of inertia of the portionof the parabola f/2 4a# bounded by the axis and the latus rectum, about the #-axis supposing the surface-density at each pointto vary the nth power of the abscissa. as Here the element of mass is =

/x being a

constant,and the Lt

where

moment

2/i#V*"a? 8y

the limits for y

are

from

or

0 to

of inertia is //,\

\y*xndxdy,

2vW, and

for

x

from

0 to

a.

INTEGRATION.

DOUBLE

We

thus

201

get

Mom.

In.

"

=

=

3

" f* oj

o

U

3

Again,the

of this

mass

+

fo

portionof

ny\/ax

the

parabolais givenby Ca\~

~~l

l*xndxdy p\ \y\ =

xndx

--

271 + 3

Thus

have

we

In. about

Mom.

0ff=3

EXAMPLES. the the first quadrant of the circle ,272+^2=a2 densityvaries at each pointas xy. Find of the quadrant, (i.)the mass " its of gravity, centre (ii.) .("") its moment of inertia about the #-axis. (iii.) 1. In

2. Work

out

the

results corresponding

parabolay^=^ax bounded by the varying as xpyq. surface-density

for the

surface

portionof

the

the latus rectum, the

axis and

varies centroid of a rod of which the line-density the distance from one end. as Find also the moments of inertia of this rod about each end and about the middle point. 3. Find

the

4. Find

the

centroid of the trianglebounded by the lines at each the surface-density y mx, x a, and the #-axis,when from varies the the of distance the as point origin. square Also the moment of inertia about the #-axis. =

=

168. For of

Polar

Curve.

polarcurves

it is desirable to

second-order Let OP, OQ be two area

curve

a

r=/(0);

Ox

Element.

Second-order use

for

our

element

infinitesimal of different form. contiguousradii vectores of the

the initial line.

Let

0, 9 + SO be

INTEGRAL

202

the

CALCULUS.

angular coordinates arcs RU, ST, with and let SO respectively, the first order. area

a

RSTU=

to this order

of rectangle

Q.

and

Draw

cular cir-

two

0 and radii r and r + Sr of Sr be small quantities

centre

and

Then sector ORU

sector OST-

=

and

of P

r

"9 Sr to the second

RSTU

sides Sr

(RS)

be considered

therefore

may

and

rSO

order,

(arcRU\

Fig,48.

Thus

if the

at each pointR(r,9) is surface-density is (tosecondcr "f)(r, 9),the mass of the element RSTU order quantities the mass of the sector err S9 Sr, and =

is therefore

Ltdr==Q[2o-rSr]S9, the

summation

being for

from

all elements

r

=

0

to

r=f(9),i.e.

"rrdr\80, Q/(0) -i

in which

to be

and

the

9 is integration taking the limit of

infinitesimal values

regardedas constant,

sum

of S9 between

of the any

sectors

for

radii specified

CALCULUS.

INTEGRAL

204

0

=2/

(Art.164)

or

169.

Centroid.

C~% rZacoaO

/

pr*dOdr

Polars.

distance of the centroid of line may be found as before

The

sectorial

a

area

from

by findingthe sum any of the moments of the elementary masses about that line and dividing of the masses. by the sum Thus its

err

SO Sr

its moment abscissa, r cos

of

element

beingthe

the

6

SO ST.

a-r

r cos

0

y-axisis

about .

and

mass

rcos(9. a-rdOdr JJ

Thus

x

j\o-rdOdr r

and

similarly

o-rdOdr

sin 0

'~fl a-r

dO dr

half of the the centroid of the upper of Art. 168. example established the result for that semi-circle that We Ex.

1.

Find

circle in

the

Also

between

/*frcos

the

limits

^or c^^ dr=

r=0

and

r

=

0 for r, and 0=0

2acos

Tfji 0R" cos

53

"S

^^

15

'

to

INTEGRATION.

DOUBLE

and

I

I jnsin0

/rsm6crrd6dr=

J J

dO

-

L4

JQ

205

Jo

/~2

sin 0 cos40 dO

Ex.

Find

2.

cardioide

the

centroid

of

the

area

bounded

by

the

uniform.

being r=a(l+cos #),the surface-density

Fig. 50. centroid is The abscissa we have

evidentlyon

the

initial line.

To

find

its

/ Ir cos 0 rdOdr "x"

the

limits for

from The

0 to

TT

r

(and

numerator

being

from

'rdOdr r=0

double, to take =2

r=a(l+cos

0),and

for 0

half).

'"

fT-1 J

-

to

in the lower

cos0o?0

L3J0

cos20 0 1a3 /"'(cos + 3

+ 3 cos30 +

cos*0)dO

CALCULUS.

INTEGRAL

206

=

| T(3cos2"9 a3

cos46

+

1 ^4--

; '

'

'

4

o3;r

2

2

5

5

'

-

4

2

'

-

2

r~r2" Ia(l+cos 6)

/TT

dO

L2Jo

0

T(l+2

a2

=

cos

0+cos20)dO

0

rf r

2a2/ (l+ cos26")d"9

Hence

x

=

-?ra3

/-?ra2

=

-a.

varies as the nfh power Ex. 3. In a circle the surface-density 0 of the distance from a point on the circumference. Find the about an axis through 0 perpenof inertia of the area moment dicular to the planeof the circle. the Here, taking0 for originand the diameter for initial line, radius. is the r=2acos a The density 6, being bounding curve

=p,r". Hence moment

Hence

of the element and its rSOSr is //,rn+1S$Sr, axis is //,rn+38$ of inertia about the specified 8r. of inertia of the disc is the moment the

mass

f ffj where

the limits for

r

0 to 2a

are

cos

0, and

for

0, 0

double). Thus

Mom.

Inertia

=

rcos"+4(9 dO J?^L(2a)"+4 4

n

+

J

to

~

(and

INTEGRATION.

DOUBLE

Again, the

of the disc is

mass

/*2acos0

r"5"

2|J ^o

=

frcosw+20d0. _?^L(2a)w+2

=

+ 2

n

Hence

207

Inertia

Mom.

Jn

4

=

EXAMPLES. 1. Find

(a) when (ft)when

of

of the sector

the centroid

circle

a

is uniform, surface-density

the

varies surface-density

the

distance

the

as

from

the centre. the centroid of a circle whose the nth power of the distance from a

varies surface-density

2. Find as

Also

point 0

on

the circumference.

of inertia

its moments

(1) about (2)about

the tangent at 0, the diameter through 0.

of uniform of inertia of the triangle that the moment the a nd lines bounded the ?/-axis by surface-density mlx+cl^ y about is the #-axis, y=mtfc + c"ft 3. Show

=

Ml 6

where

M

4. Find

is the the

the coordinate

; and

axes

triangle,they are the placedat the mid-pointsof

bounded

that

the

2

2

respectively ---

and perpendicularto of the ellipse.

'

4

~,

its

as

triangleof

that

if M

those

of

be the

uniform

of

mass

equal masses

"

the sides. of inertia

1 about

and

4

show

same

moments

_--f fC" a2 62

by

m2J

triangle.

the

5. Show

"

of inertia of the by the lines

moments

bounded surface-density

about

\ml

of the

mass

V

GI-%

and

the

of

a

major and

about

a 2

plane,M"

line

uniform minor

ellipse axes

through the

are

centre

7)2

I

M

"

,

being the

mass

INTEGRAL

208

6.

Find

the

assuming from

between

area

the

surface-density

a

the

CALCULUS.

pole,

varying

(1)

the

centroid,

(2)

the

moment

Find

of

(1)

the

for

the

r=2acos#;

r=a,

inversely

and

the

as

distance

find

inertia

perpendicular 7.

circles

about

the

to

coordinates

of

the

through

pole

plane.

included

area

line

a

between

its

the

centroid

curves

(assuming

uniform

a

surface-density), (2) (3)

8.

about

the the

Find a

line for

(2)

for

pole

the

#-axis,

this

inertia

of

the

about

revolves

area

lemniscate

perpendicular

r2

its

to

the

plane

surface-density,

surface-density

distance 9.

the

uniform a

when

of

moment

through a

about

inertia

formed

volume

the

(1)

of

moment

from

varying

the

the

as

of

square

the

pole.

Find

(1)

the

coordinates

of

the

#=a(0-hsin$), (2)

the

volume

formed

of

centroid

y=a(l by

its

"

the

area

cos0)

revolution

(a)

about

the

base

(y=2a),

8

about

the

axis

(#=0),

about

the

tangent

at

the

vertex.

;

of

the

cycloid

ELEMENTARY

DIFFERENTIAL

EQUATIONS.

E.

I.

C,

DIFFERENTIAL

212

EQUATIONS. for the

and

definite value, the same different for different curves

of the

same

but

curve

family.

Problems

in which it is necessary occur frequently to treat the whole family of curves as, for together, each instance,in findinganother family of curves,

member set at

of which a

that

letter

ought

a

functions one

rightangle. And it will be the particularizing operations,

givenangle, say

manifest

to be

of the former

intersects each member a

for such to

not

as

appear

operatedupon,

individual

or

of the

curve

should

we

in the

constant

a

system

be treating

instead

of the

whole

familycollectively. Now be got rid of thus : a may Solve for a ; we then put the equationinto "

0(",y)

=

and

a,

the form

(2)

........................

differentiation with

regard to x, a goes out, and an equation involving x, y and yv replaces equation(1). This is then the differential equation to the family of curves, of which equation(1)is the typical equation of

upon

member.

a

In the formation

be

of the differential solve

to impracticable

case

we

differentiate the

for the

Q

=

and thus

respectto

then

x

obtaininga

example,consider values to giving special For

between

a

(1)

'

........

.............

equations(1) and (3),

relation between

is true for the whole

this

obtain

and

eliminate

In

constant.

equation

f(x,y,a) with

equationit may

x, y, and

yv

which

family. the the

family of straightlines obtained by arbitraryconstant in the equation

ORDER

Solvingfor and

OF

EQ UA TION.

AN

213

w,

differentiating, Blether-wise, first solving for m, we have

or

y=

without

yi and

therefore

This

then

=

m,

y=%yiis the

differential

passingthrough the origin,and

equationof

all

straightlines

the obvious geometrical that fact that the direction of the straightline is the same as line. of the vector from the originat all pointsof the same

172. Again, suppose the the familyof curves to be

expresses

representative equationof

ftx9y,a,b) 0, arbitraryconstants a,

(1)

=

b whose values containingtwo of the family. A the several members particularize singledifferentiation with regard to x will result in a relation connecting x, y, yv a, b ; say

4(x9y9yl9a9b)0 differentiate againwith regardto

(2)

=

If

we

obtain

a

relation

connectingx,

\MX and

will

the

V" 2/i"2/2"a"

x

we

shall

y, yv y2, a, b ; say

") "" =

(3)

cally equationsa and b may theoretiappeared be eliminated (if they have not alreadydisand there by the process of differentiation), result a relation connecting x, y, yv y2 ; say

from

these three

="" F(x"y,yi"yz) differential equationof the family.

173. Order

of

an

Equation.

We define the order of a differential equationto be the order of the highestdifferential coefficientoccurring in it ; and

we

have

seen

that

if

an

equationbetween

DIFFERENTIAL

214

EQUATIONS.

unknowns contains one arbitraryconstant the result of eliminating that constant is a differential equation of the firstorder; and if it contain two two

the result is

constants arbitrary

of the second that

so

differential equation

: argument is general arbitraryconstants we shall

n

our

and the result is proceedto n differentiations, an(l differential equationconnecting x, y, yv ...,2/n"

have a

And

order. to eliminate

a

to

is therefore of the nth Ex.

1.

Eliminate

a

and

order. c

from

the

equationx2+y2=2ax +

x -f yy^ a. Differentiating, Differentiating again,l+^+y^^^ and the constants having disappearedwe

c.

=

eliminant

have

obtained

as

their

differential equation of the second order (?/2 being differential w hich the highest coefficient involved), belongsto all a

circles whose

centres

lie on

the #-axis.

the differential equationof all central conies Form whose coincide with the axes of coordinates. axes the typicalequation of a member of this family of Here Ex.

and

we

2.

have

and whence is the differential

x(y? + yy2)-yyl=0 equationsought.

174. Elimination

an

irreversible process.

Now

this process of elimination is not in generala reversible process, and when wish to discover the we of a family of curves typicalequationof a member when

a

the differential equationis given,we are pelled comof integration, to fall back, as in the case upon

set of standard

which We

are

cases,

and

many

equations may

arise

not solvable at all.

however, that in attemptingto solve may infer, differential equation of the nth order we to a are search for an algebraical relation between x, y, and n

VARIABLES

SEPARABLE.

215

these constants arbitraryconstants, such that when eliminated the given differential equation will are result. Such is regarded as the solution most a general solution obtainable. DIFFERENTIAL There

175. CASE All

EQUATIONS

equationsin

other,

x's to

one

under

come

FIRST

THE

five standard

are

I. Variables

all the

OF

ORDER.

forms.

Separable. it is

possibleto get dx and side,and dy and all the y's to the this head, and solve immediately which

by integration. Thus

Ex.1,

if

sec

x-",

sec

y=

dx have

we

cos

and a

sin

integrating, relation containingone Ex.

"

x

=

cos

y

sin y + A

x =

(x

have

+

x

=

containingone

32

+

"

(y2+ y)dy,

J

2

therefore

A.

dx

)dx

-

\

and

,

xy-^*

y+l we

dy,

arbitraryconstant

If

2.

dx

x

logx"^

+y~+ A,

2

32

arbitraryconstant

A.

EXAMPLES. Solve

y

the

followingdifferential equations :

"

1.

I

dy=x*+x+\

dy

'

dx

/

4. Show

cuts

every

*++l' that member

y*+y+l

'

dxx*+x+l every

member

of the

of the set in Ex.

family of curves right angles.

2 at

in Ex.

3

DIFFERENTIAL

216 7. Show

equal

that all the

to

EQUATIONS.

for which of the radius

the square

curves

square

vector

of the normal is either circles or

are

hyperbolae. rectangular 8. Show

makes

that

a

constant

a

for which the tangent at each point angle (a)with the radius vector can belong to curve

r=Ae^

other class than

no

9. Find

cot

a.

the

for which equationsof the curves (1) the Cartesian subtangentis constant,

(2) the

Cartesian subnormal is constant, the Polar is (3) subtangent constant, the is constant. Polar subnormal (4) 10. Find

tangent

the Cartesian equationof the is of constant length.

176. CASE

II. Linear

P, Q,

.

.

.

,

K, R

the

Equations.

[DEF. An equationof when

for which

curve

the form

functions

are

of

x

constants

or

is

lies in the fact that said to be linear. Its peculiarity differential coefficient occurs raised to a power no higherthan the first.] As we are now equationsof the first discussing

order,we

limited for the

are

that

throughoutby multiplied

we

er

case

it will be

write it

may

d ,

dPe

/Pefccv n

)="^fPdx

"

yefPdx=\Qe/Pdxdx+

Thus a

the

' *

If this be seen

presentto

relation

between

x

differential equation,and It is therefore constant. The

factor

of the called

"

an

'

e

the given satisfying containingan arbitrary the solution required.

and

y

the left-hand member perfectdifferential coefficient is

which

rendered

equationa factor." integrating

Ex.

Integrateyi+xy

1 .

fxd~

e-**

Here

217

EQUATIONS.

LINEAR

x.

=

-

e2 is

or

integratingfactor,and

an

the

equation

be written

mav

d

-

-

(ye*)=xe*, ax *2

"

ye*=e*+A,

or

+

i.e.

y

Ex.

Integrate

2.

Here

the

=

^l +

-y=x2.

dx

x

factor integrating

2.

l+Ae

is e

Jldx =elogx=x,and x

the

equation

be written

may

*JH"-+ x*

and

xy=--+A,

177.

Equations reducible

equations,if

Many

not

to

linear

form.

immediatelyof

the

linear

form

_

be immediatelyreduced may variables. One

of the

most

to it

important cases

equation

y-n

Or

Putting we

have

yl~n

=

y-^dy=^

z,

by change of is that

the

of the

DIFFERENTIAL

218

EQUATIONS.

or

is linear,and

which

(l-ri)fpdx

,~

l-n

(\-ri)fPdx

,.,

=(1

e

y

\fr\ Q-~n)fPdxJ

=(1" ft) \Qe

ze

i.e.

its solution is

x-,

dx+A. A

Integrate-^ + ^=?/2.

Ex.1,

^-2^+^1;

Here

=

putting

or

f^ (l-ri))

v

We ri)

"

dx+A,

-=0, t7

dx and

the

factor being integrating -fix*

ej*

=e-loex

^(^=-1,

have

we

x

dx\x)

x

?=logi

i.e.

X

i.e.

X

-=Ax y

"

+ x sin 2y Integratethe equation-jf.

Ex. 2.

=

Cv"^7

Dividingby cos2ywe

have

ec2y-^ + 2#

tan

y"x^.

dx

Putting we

tany=2,

^

have

+

2^=^,

GW?

and

the

factor integrating

is

"J2xdxOr e*z, giving

DIFFERENTIAL

220

Find

16.

the

of

sum

the

nth 17.

the

polar

the

radius of

power

the

as

belong

to

the

and

vector

radius

the

of

whose

the the

family polar

of

for

curves

subnormal

varies

for the

which

the

perpendicular

pedal

equation

is

radius

of

curvature

the

upon

r2-p2=^

+ "^

Jc

being 18.

a

given

Integrate

constant

the

which

vector.

curves

square class

of

equation

the

that

Show

varies

EQUATIONS.

and

equations

A

arbitrary.

%* *

normal

as

XIV.

CHAPTER

OF

EQUATIONS

HOMOGENEOUS

CASE

178.

Equations

III

ORDER"

FIRST

THE

EQUATIONS.

ONE

CLAIRAUT'S

FORM.

LETTER

ABSENT.

Equations.

Homogeneous in

homogeneous

CONTINUED.

and

x

y

may

be

written

dx

(a)

In

this

a

result

obtain

case

of

the

y

obtain

v

+

x^

dv

comes

the

variables under

Case

are

for

-^, and

vxy

=

"j"(v)"

=

_dx

~"p(v)"v and

possible

form

Putting we

if

solve

we

'

x

separated

I., giving log

result

as

r 1

Ax "

and

the

solution

thus

DIFFERENTIAL

222

if it be

(6) But solve

for

-". we

EQUATIONS.

inconvenient

solve for

or

and

"

to impracticable

write

.

dx

x

p

for

and

"-

dx

.

have

we

y

x"f"(p)

=

...............................

with respectto Differentiating

(1)

x,

dx="["'(p)dp

or

x

-"

this equationwe have x expressedas a Integrating function of p and an arbitrary constant (2) Ax=F(p)(**y) Eliminatingp between equations(1) and (2) we obtain the solution required. .........................

Ex.

1.

Solve

(x*+y*)ty-=xy. dx

and

putting

y=v%,

^+v dx dv or

x"

=

-

dx

or

og=-2 a;2

Ay^eW.

or

Ex.

2.

Suppose the equation to x

dx

\dx)

be

'

HOMOGENEO

Then

p

US

=

(p +p2)

UA

EQ

+

x(l

+

TIONS.

223

2p),

p

log J,#+2logp

giving

-=0,

"

P

i.e.

and

the

jo-eliminant

between

p2+p=" x

\

Axp*="

and is the

solution

sought.

This

eliminant

But

when

is

it of

elimination

"

algebraically impossible to if performed, the when,

is or

p,

to leave manifestly unwieldy, it is customary to regard them containing p unaltered, and would equations whose jo-eliminant if found

solution.

EXAMPLES. Solve

the

differential

.=.

.

dx

2.

equations

x+y

the

perform result two

will

the be

equations

as

simultaneous

be

the

required

DIFFERENTIAL

224

179. A The

EQUATIONS.

Special Case.

equation

^-

-

~

is readilyreduced

-

"f

"

,

dx ax+oy+c the homogeneousform thus : Put x

to

"

bk +

a^+ by+ (ah+

^

TVi

c)

"

'

dg-a'g+b'' Now

choose h, k

that

so

.1

^e.

so

that

=

"

r-/ oc

"be^-

-

"

-"

,-

ca

^

-^ ao

"ca

"

a

6

^=

Then

equation being homogeneous we may variables an(i ^ne are as separable put n~v^ This

now

before

shown. is 180. There cannot be chosen

however, in which above, viz.,when

one

case,

as

a

b

c

6'

c''

_ ~~

a' Now

let

and

=m

"

a

dy

Then

so

that

Tx= n -~

--

7

"

a "*

= -

-

V

dx

my

drj (am

-

/)

+

c

+

+ ad b)rj

+ 6c

_

dec"" and

mrj +

c

-

-, ,

,

".

.

-,

+bc (am-\-o)r)-\-ac

n.

h, k

HOMOGENEOUS

The

EQUATIONS.

225

the integration beingnow separated, be at once performed. may 181. One other case is worthy of notice,viz., ax + by+ c dy dx~ "bx + b'y+ c" when the coefficient of y in the numerator is equalto that of x in the denominator with the opposite sign. For then we may write the equationthus + c')dy (ax+ c)dx+ b(ydx+x dy) (b'y differential equation exact an being ; the integral ax2 + 2cx + 2bxy b'y2 + 2c'y + A A. beingthe arbitrary constant. variables

_

=

"

"

=

,

Integrate =y-

Ex.l.

dx

Put

Choose

#="+

k, y

h and

Icso

=

ri+

x+y-Z k, so

that

that =

then Now

then put 77=0(1,

_

~

l+v

"-

v+1

1)2-

l

where

"=#"1

and

v=^~ .

x"\ E. I. c.

p

'

DIFFERENTIAL

226

Ex. 2.

Integratef

*+* =

dx

.

x+y

\

"

.#+y=??, then .=

..

Let

EQUATIONS.

dx

and

1

??

-

if] 1

'

"

^

where

?7=

EXAMPLES.

Integratethe equations: dy _

dx

bx+ay-b ^

"

1*

8 9. Show

that

a

particle #,

y which

moves

so

that

~

will

always lie upon

10. Show

fUL '

\"

that

~)

a

conic section.

solutions

of the

generalhomogeneous

always represent

must

families

of

tion equa-

similar

dx)

curves.

11. Show

that solutions

of

/(-,-j-} J are

\X

of y and some power the typical equationof in x, y and some

a a

homogeneous in

singleconstant, and converselythat member power

x,

CLX

of a of one

family of

curves

constant,the

if

be homogeneous differential

DIFFERENTIAL

228

Differentiate with

x, i.e.the absent

regard to

The

P

and

dx

Thus

EQUATIONS.

x

After

is performedwe integration and this equationand 2/ "/"(p) between of the givenequationis obtained.

the

=

which

eliminate p the solution

absent.

183. y B.

letter.

absent from then takes the form

the differential

Suppose y

equation,

fj1J

Since

-^

=

ax

"

this may

be written

ax

dy dx'

and

therefore if y

variable the

regardedas the independent remarks foregoing apply to this case also. be

Thus dx

(i.)if

convenient result of the form

solve

we

dx

.

,

a5T*S dx 7 dy

then

and the

=

"7-^,

is integral

dx

for

-^-, ^

and

obtain

a

ONE

LETTER

if this solution

(ii.)But

ABSENT.

for

-7-

be inconvenient

solve for

x

obtain

and

yy x

=

or

dy

we impossible

form

229

where (j)(q)

with

q stands

regardto

for

a

result of the

/v"

Then

-j-

tiating differen-

y, the absent letter,

Thus

and After

the

integrationwe and equation and x "f)(q). equationis obtained.

the

=

The or

y

student

absent,we

should

note

solve for

between solution of the

eliminate

q

that in either case,

~

x

this

given absent

if possible. by preference

ax

But

when

inconvenient solve we or impossible for the remainingletter and differentiate with regard the absent letter to the absent one\ thus considering in either case the independentvariable. as Ex.

1.

this is

Integratethe equation

'

Here

dy and

#=

"

2

is the solution. Ex.2.

Solve dx

Then

where

=

\dx) x

q=.

dy

DIFFERENTIAL

230 Then

EQUATIONS. absent lettery,

with regardto the differentiating /, 1 \dq l~

*

and

#

and

the

^-eliminantbetween

this

is the solution

equationx=q+~

equation

and

the

original

required.

EXAMPLES. Solve 1.

the

dy =

equations:

I.

y +

5.

6.

-B

\dx)

4.

(2a^ + ^2)=a2+2a^7.

\dx)

Clairaut's

V.

Writing^9 for

with Differentiating

dx

Form,

^=

have

we

-~-

=A+".

8.

dx

184. CASE

dx

y=px+f(p) regardto x,

........................

(1)

dp

{x+f(p)}0,

or

whence

either

-^-=0

or

....................

cc+//(p) 0. =

CLOu

Now

-f

=

"

givesp

=

C

a,

constant.

(2)

CLAIRA

UTS

FORM.

231

is a solution of the given differential Cx-\-f(C) C. constant equationcontainingan arbitrary Again,if p be found as a function of x from the equation (3) "+/(,) 0, Thus

y

=

=

........................

equation(2)will

stillbe satisfied, and if this value of which is the same or p be substituted in equation(1), if p be eliminated between thing, equations(1)and (3) shall obtain a relation between we y and x which also satisfiesthe differential equation Now to eliminate p between

y=px+f(p)}

x+f(p)I

0=

is the

same

as

to eliminate

0

i.e.the

same =

(2) The

x+f(0)J

=

the process of findingthe envelopeof Cx+f(C) for different values of 0. therefore two classes of solutions, viz. : as

the line y There are

(1) The

0 between

linear

called the solution,

"

completeprimitive," constant. an containing arbitrary "

solution envelopeor singular containing and constant from derivable not no arbitrary the complete primitive by putting any numerical value for the constant particular "

in that solution.

The

geometrical relation between

these

two

tions solu-

is that of a familyof lines and their envelope. It is beyond the scope of this book to discuss fully the theory of singularsolutions,and the student is referred to largertreatises for further information upon

the

subject.

EQUATIONS.

DIFFERENTIAL

232

Solve

Ex.

y=jt

JP

and m

is completeprimitive

Clairaut's rule the

By

the

is the result of

solution envelopeor singular the above

between

eliminating

equationand

o=*--2. m2 y*="ax.

i.e. The

student

y2- 4a#

the

=

will at

equation to the

y=mx+"

once

well

a

recognizein and parabola,

the singularsolution in the completeprimitive

equation of

known

a

tangent

to

the

parabola. EXAMPLES. find the

and the complete primitive, down Write cases : solution in each of the following

envelope

"

4. y" 5. y

6.

185. The

(x

=

"

a)p

"

p^.

(y"

equation y=x"P(p)+*Kp),

(i)

.....................

with regard to x, by differentiating variable. then considering p as the independent have For differentiating, we

may and

be

solved

"

whence

-=

--

dp which

is

the solution being linear,

["P(P)*P r .,,Ji eJ"t*P)-P=_ (P) xe"t*P)-P=_

tW-Pdp+

A

.......

(2)

233

EXAMPLES.

If

equations(1)and originalequation

between

p be eliminated

now

(2),the completeprimitive of the will result. Ex.

Solve

We

y

have

=

2px+p2.

(1)

,

p

.

(2) p^x" %p3 A giving be found from these two equationsmay The jo-eliminant now in equation(2). by solvingequation(1)for p, and substituting But if it be an objectto present the result in rational form, we may proceedthus : 0,\ By equation(2) 2p3+ 3p2#+ SA from (1) 0. / + Ip^x-py j93 "

"

..............................

"

=

=

Hen

And

p*x Zpy

ce

-

-

3A

=

equationand

this

between by cross-multiplication

givingas

0.

the eliminant

4(y2+ 3Ax)(x*+y)

=

(xy

-

3 A)2.

algebraic p being process of eliminating the equations(1) difficultor impossible, in many cases and (2)are often regardedas simultaneous equations but the is the solution in question whose ""-eliminant actual elimination not performed. 186. The

EXAMPLES. Solve the

equations:

1. y= ,

2. y

=

3. y= 4. y=

p

DIFFERENTIAL

234

The

8.

and

T,

OT2

PTto

is

the

the

differential that

the

the

in

the

Form

length

A

12.

is

for

and

tangent

the

as

the

as

possess

the

by

Obtain

when

which

the is

tangent

a

complete tion solu-

singular

of

area

the

of

and

which

for

curves

the,

between

intercepted

tangent Obtain

triangle

constant.

equation the

the

interpret

the

complete

solution.

singular

satisfies

that^"=0

made

of

1888.]

which

curves

intercepts

tangent,

constant.

the

curve

all

constant.

the

of

portion

and

the

differential

axes

primitive

of

and

axes

the

coordinate

is

curves

the

of

[OXFOKD, of

differential

the

x=\

;

determine

its

and

y=p\x"p))

equation

equation. 1889.]

[OXFOKD, IS.

Find

the

in

Oy

inclination

the

of

tangent

axis

question.

the

by

11.

sum

the

meets

curve

curve.

of

axes

curves

bounded

the

a

equation

equation

Obtain

10.

the

the

to

Find

coordinate

primitive

also

proportional

of

P

point

any

Ox.

property the

on

at

axis

Find

9.

the

tangent

EQUATIONS.

complete

primitive

and

singular

solution

of

the

equation

14.

is

Show

reduced Hence

solution.

that

to

write

one

by

putting

of

Clairaut's

down

Interpret

its the

x2"s

and

y2

=

[OXFORD, ", the

1890.]

equation

form.

complete result.

'

V^yj

\

dx

primitive

and

find

its

singular

DIFFERENTIAL

236 Then

yi

=

2/2

=

Thus

on

But

EQUATIONS.

zj(x)+zf(x); zj(x)+^'(x)+zf"(x).

substitution

get

we

+ Pf(x)+ Qf(x) /'(*) =

0

by hypothesis.Hence

z

an

equationwhich is linear factor is The integrating

for zv

or

is and the first integral

the second integral whence may effected. solution and the Ex.

Solve

y=x

once

L

da? Here

be at

makes

dx

-r^

Put then

Hence

**

/K" 4- 3^

and

the

factor integrating

is

e^

x

or

x*e

4 .

obtained

237

x*

j

Thus

EQUATIONS.

ORDER

SECOND

~(zlx^)=x^ a*

5

z1x2e*=~+A

and

_*

whence and

z=-\e

the solution

r

+ A

*

,

\ "-e

J a?

requiredis -=

5

189. CASE A. If

x

II

be

letter absent.

One

absent, let y1=p, -

and

the

=

"f"(y, p, p-S-\Q,

takes the form

=

dy/

\

and

0

"f"(y, yv y2)

equation

is of the first order.

B. If y be the letter

absent,let

yl =p,

*-" and

"f"(x, yv y%) becomes

and

again is of

Ex.1. Here

The

or

Solve x

the firstorder.

the

is absent.

equationyy2+#i2=2#2. So

factor integrating

puttingy"=p

is e^

p2y2"y* +

vAy or

constant

and

y2=p^?, have

y2,

=?/4+ a4,say.

we

EQUATIONS.

DIFFERENTIAL

238

Hence

or

+ A). y* a2sinh(2#

i.e.

=

l+#i2=#y2#i"

Solve

Ex. 2.

y is absent.

Here

So

puttingy-^"p^

dx or

pdp

-"--"5,

=

"

1+jtr

x

i.e.

logx

logVl+p2

=

+ constant.

^.e.

ady" ^Jx*

or

"

and

dx,

oy=i?^2_"

giving a

a?

b

constants. being arbitrary

EXAMPLES. Solve the 1. ^2

=

followingequations: "

6.

1.

2.

1+3^=^2.

7.

3.

i+y!2-^2-

8-

1

y2+"/i-y=-e

_L2 2-

^"/

4.

5. "3/2

=

10. Solve

that

a

9y22=4iyi-

^

=

y#2=#i3-#i-

[OXFORD, 1889.]

(l+.yM the

0 when

=2/" havinggiven ~#2)^~#(^)

equation(1 y

=

0.

[OXFORD, 1890. ]

ow?

11. Given

find the

that #2 is

a

value

completesolution.

of y which

satisfies the

equation

[L 0. S., 1894.]

REMOVAL

Let

Linear

General

190. Term.

Q Pv P2, Puttingy vz, we

where

.

.

.

239

Removal

of

a

generalequation

more

of

given functions

are

,

x.

have

=

y2

TERM.

A

Equation.

consider the

next

us

OF

vz2 +

=

2V"

+ v2z, etc.,

whence

n(n"

1)

, -

--

-

2

The coefficientof 0n_i is If then v be chosen so that P

dv

"

or

"

v

the term

chosen

the term containing 0W_2 The coefficient of z is

if

a

will make

zl the =

value this

=

e

of

v

been

will have

zn-i involving if be so Similarly, v equation

and

v

n

as

to

will have

the satisfy

been

be found

can

removed.

expressionvanish,we

or

ential differ-

removed.

guessedwhich can, by writing

therefore z2 zn rjn-i, reduce rjly etc.,and degree of the equationby unity. The student

rj,

should

and

notice

=

=

that

this

expressionis

the

same

in

DIFFERENTIAL

240

form

the

left hand

if any

solution

as

Hence

EQUATIONS. member y

"

v

of the

given equation. be found or guessed of is right hand member

can

the

the given equation when omitted, we by writing y can, vz, reduce the degree of the equation.

and

=

Canonical

191.

In the

the

substitution

will

by what equation to

But at

equation

y

has the

"EXACT"

p

of the

above

can

For

be

"

q.

denoting

by

of this

equation

given

has

yq,

=

not

been

EQUATION. is

xp-r~

etc.,

Thus

the

simpler form

integratedwhatever

\xPyqdx

Y\,

degree

reduce

stated

DIFFERENTIAL

is

second

an

exact

^

and

"

e~l ^dxz

sometimes

general solution present effected.

When

=

been

the

192.

Z^

Form.

of the

case

then

y may

be.

differential.

EQUATION.

DIFFERENTIAL

EXACT

It will be noticed

that when be effected.

cannot

By aid of quicklywhether 193.

the above

given

a

lemma

"

or

q =p

241

the integrati

p

may often see equation is " exact/' For we

if all terms of the form xpyq in which p is first removed, we tell at once can frequently whether the remainder coefficient or not. Ex.

is a

q be spectio by in-

"

differential perfect

#2

Here, by the lemma, #2y5 and

x?y" are

perfectdifferential

and obviouslyocy^-\-yis coefficients, of xy.

Hence

first

a

the differential coefficient of integral this differential equation is

obviously cos

=-

194. A A

more

generaltest

equationmay

Test. for

an

"exact"

be established in the

For upon

n-

differential

generalcase

forms

the coefficients P0,Pv have, providedthey be functions of x. whatever

.

General

more

x+A.

...

,

Pn, V may

denotingdifferentiations by dashes, we integration by parts

Bysdx=Pn-

32/2 -

Pn

-

8^1 +

P"n

-

zV

-

IP"'n

etc.

Hence

p. i. c.

upon

addition it is obvious

have

that if

-

EQUATIONS.

DIFFERENTIAL

242

given equation

the

exact; and

is

tegral its first in-

that

is

Is the

Ex.

and

=

PQ'" 24^ P3 P2'+ PI equationis exact ; and its

the

Thus

=

-

-

exact

x

?

have

the test,we

Applying

sin

+ 1 2x?y2 + SG^2^ + 24#?/ equationcfiyz

-

72^ + 72^ first

24^

-

0,

=

integralis

or

This

which

be

again will is satisfied.

perfectdifferential

a

Hence

a

(8#3 43%

second

-f ^4yx =

-

-

if

integralwill sin

x

+ Ax

4^73y+^4y1= sin^+^^

or

"

again be tested. may is third and final integral

which

^

But

it is

now

be

+

B,

+

J?,

obvious

that

the

cos.*?+

=

IB

EXAMPLES. 1. Show

exact, and 2. Solve

^7/3+ 3. Write

that

the

solve it the

equation

completely.

equation

.

+ sin

6^/2 +

%i

down

first

x(y* %i) + ~

of integrals

the

cos

X33/2 !/} =

-

sin

^

followingequations :"

'(a) (b) (c) 4. Show an

that

if the

integratingfactor

equation

equation P2y + P^y^ + P0"y2 //, will satisfy the //,,then =

F admits

of

differential

DIFFERENTIAL

244

EQUATIONS.

Hence

is

solution of

equation(1) containingn arbitrary constants,and is therefore the most generalsolution to be expected. No more generalsolution has been found. The portionf(x) is termed the Particular Integral the n arbitrary and the remainingpartcontaining (P.I.), constants,which is the solution when the right-hand member of the equationis replaced by zero, is called the Complementary Function (C.F.).If these two partscan be found the whole solution can be at once a

written

down

as

their

sum.

remarkable

196. Two

Cases.

in which readilyobtained. generally There

are

(1) When

two

the

cases

these solutions

Pv P2, quantities

Pn

...,

be

can

all

are

constants.

(2) When

the

equationtakes

r7n-2n/

Jn-l

ri

the form '

F"

2+-'-+^=

av a2, ..., an beingconstants and V any function of x. is readilyreducible, The solution of the second case as will be shown, to the solution of an equationcoming under

the firsthead.

EQUATION

WITH

COEFFICIENTS

CONSTANT

MENTARY COMPLE-

"

FUNCTION. 197. Let us therefore firstdetermine such an equationas + a22/n-2+... + 2/n+^i2/u-i the coefficients being constants

confine "

our

attention

to

ComplementaryFunction

the "

the solution of

^n2/

=

0,

; i.e.for the

presentwe

determination

in the first case.

(1)

.........

of

the

Y

COMPLEMENTAR

As

mn

+

245

put y Aemx, and we have a1mn-1+ a2raw-2+...+an 0

trial solution

a

FUNCTION.

=

(2)

=

.......

equationbe

Let the roots of this

774, m2, ms,

...,

mn,

then supposed(forthe present)all different,

and all solutions,

are

therefore also

+...+A (3) nem*x, A^x + A2e^x+ A3em*x is a solution containing n arbitraryconstants Av A2, An, and is the most generalto be expected. A3, y

=

......

...,

198. Two If two

Roots

Equal.

of

become

equal,say the of first the solution (3)become two terms mx m2, be regardedas a and since A^ + A^ may (A!+ J.2)e"11*, roots

equation (2)

=

singleconstant, there unity in the number (3) is no longer the expected. Let

us

this

examine

Put

is of

apparent diminution by arbitraryconstants,so that most general solution to be an

closely.

more

m1+A. ^x + A 2e(TOi+*X" 97i2 =

A

Then

h?x2

r =

~~\

Alem^x-\-A2(^x\ l+hx+-^-

+

...

rhy?

=

(Al+ A2)^x

Now

A^ and and quantities, of two

A2 we

other

+

AJi

are

may

.

~\

I. xem^x+Azhem^~^

two

+

.

..

independentarbitrary

therefore

express

them

in

independentarbitraryquantities by two relations chosen at our pleasure. First we will choose A2 so largethat ultimately small may be written "2, A2h when h is indefinitely an arbitraryfinite constant.

terms

DIFFERENTIAL

246

EQ

UA

TIONS.

will choose A1 so largeand of opposite we Secondly, signto A2 that A^+A2 may be regardedas an arbitrary finite constant Bv Then the terms

vanish with h since Aji has been considered ultimately is confinite and the expressionin square brackets vergent and Thus

h

contains

as

factor.

a

A^^+A^e11^ may, when m2 mv and therefore ultimatelyreplacedby B1emiX+B2xemiX" in the whole of arbitrary constants the number

be

the terms

=

solution remains we n, and the generalsolution in this 199. Three

Consider

therefore

obtained

have

case.

Equal Roots. the become

next

three of the roots of The equal,viz.,m1 m2 m3.

case

when

equation(2) have already been terms, Alem"+A#m*x+A2f?ri*x, by (Bi+B^e^+Atf"**. Let

ms

=

A^x

+ Ae'W

for A+Aw

Thus

and

AjPtifc A^x( =

=

we

may

so

choose

Ov C2,03 being any

placed re-

mx + h fcZftZ

/

Then

=

=

1 + kx + we

-^-

\

+...)"

have

AB,52,and Bv

that

arbitraryconstants, whatever

k

COMPLEMENTARY

FUNCTION.

be, providedit be

may

finite

absolute

not

247 But

zero.

AJc2

series within being chosen a the square brackets beingconvergent, it is clear that when k is indefinitely diminished, the ultimately, form limiting

of this

200.

Several

In

similar

a

roots of the

and quantity,

is expression

Roots

Equal. it will be

manner

equation(2)become m^ m2 loss of =

there

= .

.

that

if p

equal,viz., =

.

obvious

mp,

in our solution if generality expression + KjxP *)"*", (%!+ K^x + Kfl?+ portionof the complementary corresponding

will be no substitute the

we

the

-

.

for the

.

.

function,viz.,

A^x

+

+...+ Apew*"x. A2em*x

201. Generalization. More

if generally,

be the

ential complementaryfunction of any linear differequationwith or without constant coefficients, what is to replace this expression to re-tain the so as when generality mx m2 ? =

Let

m2

Then

and the terms

become + A2"p(m2) Al"f"(m1) h2

Now two

putting A^A^ Bly AJi "2) arbitraryfinite constants,the remainingterms =

=

DIFFERENTIAL

248

EQUATIONS.

ultimately disappearwhen we approachthe limit diminished. which h is indefinitely Thus Al"f)(ml)+ J.20(m2)may be replaced by

in

number (n) of arbitrary retainingthe same constants An B1952,A2, A^ in the complementary function as it originally possessed. And as in Art. 200 we proceedto show that if may viz. ^i1 m2= p roots become equal, =mp, the terms + 420(m2)+ +Ap"j"(mp) J.10(ra1) be replaced by may thus

...,

=

...

"

when

the

of generality

.

.

the solution will be retained.

results of Arts. 198, 199, 200 are of course ticular parof this,the form of "p(/m^) emix. cases being

The

202.

Imaginary Roots.

When

a

root of

equation(2)of

Art. 197 is imaginary,

it is to be remembered that for equationswith coefficients imaginaryroots occur in pairs.

Suppose,for instance,we where

i

Then

=

\/

"

have

1.

the terms

A^+A^e"** may

real

be thrown

into

real form

a

(Al +

thus

: "

bx bx)+ A2eax(cos bx bx + (A1 ^2"6a*sin A2)eaxeos 1

=

A^

or

sin

"

-

sm

bx,

i

sin

bx)

FUNCTION.

COMPLEMENTARY

where

249

constants Bl and B2 replace arbitrary (Al A2)irespectively. then cos a, B% p sin a,

the two

A^+ A2 and Let B^ p =

"

=

JB*

=

Then

+

"22

and

-

ijgF.

"2sinbx p cos(bx a). We may thus further replace bx by CLeaa!cos(6aj jB^cos bx + B2eaxsin constants. where C^ and 02 are arbitrary ^cos

203.

fr#+

tan

=

a

=

Repeated Imaginary

"

Roots.

For

repeatedimaginary roots we may proceedas that when before,for it has been shown 7772 ??i1, Alem^x+A^x may be replacedby (J^+J?^***, and if m4 by m3, A^X+A^X may be replaced =

=

If then mx

=

m2

=

a

+ ib and

m3

"

m4

=

a

"

f

6,we

may

replace by that is

(

by

sin te] bx + (Bl- B3)i + 53)cos eax[(Bl + xeax[(B2 + ^4)cos60?+ (B2 "4)^sin 6 and therefore by 6^+ (72sin 6x+ (74sin +cceaaj((73cos e^CC^cos 6aj) that is by 6aj+ ^ + cc(73)cos tf*(Ci which is the same or thingby -

Any

of the last three

constants constants

which

forms

replacethe Av A2, A^ A^ and

contain

four

originalfour thus

arbitrary arbitrary

retain intact

the

DIFFERENTIAL

250

EQUATIONS.

(n) of arbitraryconstants requisite proper number to make the whole solution the most generalto be expected. And this rule may obviouslybe extended to the case when of the imaginaryroots any number equal.

are

204. Ex.

1.

Solve

equation^-

the

dx

dx2

Here

whose

trial solution

our

roots

are

Accordinglyy

1 and

is y=Aemxy and

obtain

2.

A^e*and

"

we

and

y

=

A2e2xare

both

solutions, particular

y=A1e*+A2e2x

is the

constants. generalsolution containingtwo arbitrary

Ex.2.

Solve

-*V-a?y=0. aOC

Here the auxiliary equation is w2 and the generalsolution is

or

as

it may

be written y

by replacingAl by

"

a2

ax .Z^cosh

=

Bi+B*

and

+

^sinh

"a,

A2 by

ax

B^~B 2

the auxiliaryequationis m2-f-a2=0 the general solution is y

or, which

is its

Ex.4

Solve

=

=

stands

+

roots

m"

+ai.

^t2si

Bl

?-4| ax?

D

AjCosax

with

equivalent, y

where

m"

Solve

Ex.3.

Hence

roots

(ifdesired)

2

Here

0 with

=

for

ax

"

.

ax

+

5-2y ax

=

0

or

(D- l)2("-2)y 0, =

3-

4-

5.

6.

EQUATIONS.

DIFFERENTIAL

252

S-9 S~3 g=y. g=y.

9.

10.

("-

11. 12.

THE

of such

a2,

a

=

beingconstants,and Fany

an

...,

next

we

INTEGRAL.

Having considered the complementaryfunction an equationas F(D)y V where F(D) stands for

205.

av

PARTICULAR

turn

our

We

most

propose useful of the processes above

the

write

may

of x,

of

to the mode

attention

and particular integral,

and

function

obtaining to givethe ordinary adopted.

equation as

2/

=

1

where (or [/(D)]F),

satisfies

206. "Z""

is such

^7^

the

operatorthat

an

fundamental

laws

of

that

the

Algebra. It is shown

in

the

Differential

Calculus

satisfies (denoting -y- j

operatorD

(1) The

Distributive

(2) The

Commutative

i.e.

Law

of

Law

D(cu)

=

Algebra,viz. as

far

c(Du}.

as

stants, regards con-

INTEGRAL.

PARTICULAR

(3) The and

Index

253

Law, i.e.

integers. beingpositive Thus the symbol D satisfiesall the elementaryrules the with of algebraical of combination quantities with regard to exceptionthat it is not commutative m

n

variables. It therefore

identityhas a analogue. Thus

rational

algebraical correspondingsymbolicaloperative since by the binomial theorem

follows

that

any

'

7? l

+

(T)

~Li

"""

\JL

2i .

have

we

may

by

an

be inferred without

Let

proved in the positive integer,

a

us

define the

"

"

Differential Calculus that

D~r operation DrD~ru

Then

.

Operationf(D)eax.

It has been if r be

operatorswhich proof

further

JL

207.

for

analogoustheorem

=

to be such that

u.

D~l

and we shall representsan integration, D~lu no constants arbitrary suppose that in the operation added (forour is to obtain a are objectnow and not the most generalintegral). particular integral Now since Dra~reax eax DrD-reax,it follows that =

=

D~reax

=

a-reax.

Hence it is clear that Dneax aneax values of n positive or negative. =

for all

integral

DIFFERENTIAL

254

EQUATIONS.

pansion f(z) be any function of z capableof exof z, positive in integral or negative powers ^Arzr say, Ar beinga constant,independentof z).

208. Let

(

=

Then

result of the

The

be obtained Ex. 1.

f(D)eaxmay operation D by a. by replaci'ng

Obtain the value of

-

"

"

"

therefore

-e

^

the rule this is

Obviously by

"*

Obtain

the value

E,

2,

By

the rule this is

or

g.

of

e3a/'=-^-e

EXAMPLES. 1. Perform

the

indicated by operations

'

^

3.

Apply

Art. 208

to show

that

m^7 /(Z)2)sin mx /(Z)2)cos

=/( =/(

209.

Operation f(D)eaxX.

Next

let y

=

eaxY,where

"

"

Fis

m2)si m2)c

any

function

of

x.

INTEGRAL.

PARTICULAR

have

we

Dreax

since

Then

255

areax,

=

Leibnitz's Theorem

by

1D Y+ nC2D2Y+...+Dn F+ n(71an edx(an -

yn

=

which, by analogy with 206), may be written

the

Dneax Y= n

Binomial

Theorem

F), (Art,

eax(D+ a)nF,

beinga positive integer. Now that

so

X

let we

from

Then

write

may

above

DneaxY=eax(D+a)nY nX

Dneax(D+ a)

~

or

and

therefore

Hence

in all

=

eaxX,

D for

cases

integralvalues

of

n

or positive

negative DneaxX 210. As

in Art. 208

eax(D+ a)nX.

=

shall have

we

f(D)eaxX

=

be

That is,eax may the leftof the by D + a.

Ex. 2.

"

-

D2

"

"

"

transferred from the rightside to operator f(D) provided we replaceD

-

-

4D + 4

e2xsinx

=

e2x~ D

sin *

x"

"

e~xsin x.

DIFFERENTIAL

256

EQUATIONS.

EXAMPLES. 1. Perform

the

operations

1

(D

If*

-

1 l

h

(D-I)*6

'

D-l

Operation/(7"2)

211.

sn

D2

have

wwu

(

=

m2)y

-

cos

and

X'

that

2. Show

We

1

o

X

sin mx, cos

therefore

Hence,

before,Arts.

as

a^ j;/ r"9\

n

J

Ex.

D

^

sin

/v

) '

feaxsin bx dx

and

208

mx

y

"

cos

6^ (Art.210) eax(D+ a)~1sin

=

-_

mx.

"

Z)-1eaxsin 6^7

7)Wri

hv

f'Arf 91^ xxi li, Zi L 1

X/^iSJlllf"x/

sin ".#

a

sin

o\

f( m2)

=

cos

=

210, it will follow

6

"

cos

^

bx

tan-1- Y ^sin^.r

ea*(a?+62)

-

EXAMPLES. of integrals eaxcosbx, e^sin2^,e^sin3^,si the operations Perform

1. Find

2.

by

this method

-sin 3. Obtain

by

means

the

2^,

-_

sin 2^7.

values of the sine exponential mx. /(/")cos mx, /(Z")sin operations

of the

cosine the results of the

_"I L_cos,r,

and

y

INTEGRAL.

PARTICULAR

Operation

212. Let

257

consider the

next

us

operation

F(z)is a function of z capableof expansionin integral positive powers of z. Let F(D) be arrangedin powers of Z),then if no odd the result may be written down occur by the powers rule of Art. 211. foregoing where

Thus ^ sin S1T1 2#

""

sin %x"

=

"

But

if both

follows

proceed

as

and the

and

even

the odd

odd

Group and together,

"

powers

"

sin 2#.

51 occur

powers the

:

" ~

+ 16-64

L-4

we

may

powers together then we may write

even

operation sm

mx

./T^ox

^

.

mx

sin

=

/TV,V

^

^ "

Upon we

may

m2)sinmx

mx(

"

"

m2)cosmx

that in practice it will be seen examination write m2 for Z)2 immediatelyafter the step "

writingimmediately 1 "

E. I. C.

"

"\

f)~7

^ R

sin

mx"

EQUATIONS.

DIFFERENTIAL

258

"

mi

"

in/

1.

"

"

i

J-'Xv

in

i

Obtain

of

the value

.

"

"

"

,

ma;, etc.

4^0 sin

"

"

Ex.

)

SYT^ fjsrS

r-y^

or'

-

="

sin 2#.

^

Thisis sin D

"sin

2#, 2^,

-

J.O cos

2^

the value

of

^

or

Ex.

This

2.

Obtain

expression

^

"

-^e2*cos ^p.

"

^

e2*-^

=

sin 2^7.

"

"

cos

-

x

^

each [replacing

Z"2 by

1]

"

e2*

1

____

=

"

"(cos

x

"

sin

x).

4

EXAMPLES. 1. Perform

the

operationsindicated

in

the

pressions followingex-

:-

D

Z"3

-e*sin

x

+

EQUATIONS.

DIFFERENTIAL

260

EXAMPLES. the

Perform

operations

CD+l)(Z"

2)

+

2-

3.

214.

J?

-

Cases

COsh

COS

37

07.

Failure.

of

applying the above Particular Integral, cases with.

met

We

procedure 215.

Ex.

be

to

adopted

illustrate

in such

To

If

we

208, the

Art.

apply

Integral

Particular

the

obtain

result

i^i We

this

evade

may

operation by applying

210

'

have

we

becomes

or

""-

obtain

difficultyand Art.

when

we

the

the

result

of

the

have

particular integral required. another of substituting method, however, Instead,

which

of

course

is Ae*.

Function

Complementary

the

cases.

dx The

are

obtaining a frequently

(^L-y=ex.

equation

the

Solve

1.

failure

of to

propose

of

methods

In

is the

operation

Writing

=

x(\ +

"

-"

h)

more

instead

carefully. of #,

we

have

let

us

examine

263 OF

CASES

expressionthe portionLtex//ibecomes taken with the complementaryfunction Aex

infinite,

this

Of

be

may

FAILURE.

arbitrarywe

regard A

may

+

; and

arbitraryconstan

as

a

new

to contain

a

negativelyinfinite

-

b

A

/i

for

we

A

suppose

may

the term I/A. xe* is the Particular term

por

to cancel

The

remainingterms indefinitely.

the

Solve

Ex.2.

y

and

In

sin 2#.

211, we

We This

is

and

oo

2#.

cos

parts "i

of two

part, if

"

e*

apply the

we

o

ri

fail.

so

,

when limit,

the

consider

now

this second

2HL",i.e.

get

clearly

A sin Zx + B

=

integralconsists particular

The

Art.

^ Ct/X

equation

complementary function

The

h is decre;

is

solution

whole

when

vanish

contain h and

The

The

Integraldesired.

h

Q, of

=

--

sin2.i'(l-

"

expression 1

_1

=

4

i_(i+A)a 1

1

I

JT2^U^X

9A

COS

"^IX "*"COS

^^ S^n

^^J?)

-

1 sin 2.2? " "

l

fi

o

=

(a

"

term

1

which

may x

-

the whole

,

ofr A

4

function) Thus

2.27+ powers

cos

-x

solution

be included ^

c^s +

in the

(terms which

of the. differential

complemvanish

v

equation

g JOS

"

"

DIFFERENTIAL

260

:.

the

equation (D2+ 3D)(D -l)2y e* + complementary function

Solve

3.

EQUATIONS.

=

'ere the

integralconsists particular

e~x + sin

is

=

parts,viz.,

"?

x_

iy*6 ~(D-I*'

(a part going into +"

ex +

the

4- x1.

plainly

of four 1

1

x

=

*_ JL

T

'I?''

4~4

complementary function)

(termswhich

vanish

with

,^-PP,

10

3-Z)

1 sin

-

x=

6 + 2Z) =

(3 sin

#

"/

2(9 -

cos

-

^)/20.

jjinally

open

44

ie

y

the whole =

solution

is

Al + A2e~3x+ (A3 e2x

3 sin

x

"

cos

x

5#2

.3? .

+

~~

+'+""l"

44 ,

h)].

Ex.

4.

The

C.F.

To find the which

the

Solve

EXAMPLES.

TIVE

USTRA

ILL

263

equation"^-?/ ^sin^. =

is

P.I.

we

have

."-a

is the coefficient of

i

in 1

"?".in

#*,

l. "

"*"

plX

rp

-4^-6/^Tr. l **

Thus

the

l

~^l-^D..\^

COSJP

is

P.I.

_

8

and y

the whole =

A

^inh ^

solution +

'

3^ gin ^ 8

is

^2coshx

+

A3sm

x+A

x

4cos

x

+

c^s^' ^ -

sin

^p.

EXAMPLES. 1. Obtain

the Particular

0)

sina7'

7TTT

indicated by Integrals

(5" /n (6)

nT-T8*1111*'

_

(7) /na

w^ovn-Q-x*'(sinh#

^/

n

+ sin

^?).

^o,(^ COS

+ cosh

-

o

COS

6^).

.

o

DIFFERENTIAL

264 Solve

2.

EQUATIONS.

differential

the

(3)

+y

equations

=

Cfc#

(5) (Z)-

(4) (D*-l)(D*-l)y=xe*. (6) (ZP-3D2-3D

I)y

+

=

e-x

(8) (Z"2

(7) OD3-%=#sin.". (9) (Z"2 (10) (^"-

216.

The

Operator

""-. CvQC

reducing an

where the

equation of

Av A2,

coefficients

constants, to a form constants, arises from

this

case

are

=

-TT

therefore

e*,and

the

that

therefore

x~ax

-ji

putting

=

-^at

operators d

d are

for

stand

D

Let

equivalent.

-j-.

have

-*-

dx\

)X

n

_

n nf\n

Z. _

dxn

" "

(x \

all

et.

=

at

It is obvious

in which

are

...,

in

the class

x

In

peculiarservice

renders

which

transformation

A

__

dx

11

-I-1 ]X /

. ~

1 ___

dxn~l

x-j-

and

dx

Then

we

EXAMPLES.

Now

265

in succession

putting11

2, 3, 4, ...,

we

have

etc. ,

Hence

or

generally

the order of the operations reversing D(D-l\D-Z)...(D=

Ex.

the differential

Solve

Putting x

=

equation

c?,the equation becomes

D(D -l)(D- 2)y+ 2D(D

%

-

+

3%

-

3#

=

(

or

i.e.

(D

giving y

=

Ae* + B

-

cos

-"

,

~+;rIog EXAMPLES. Solve

the differential 1.

s 2

2.

equations

x-

dx

+ --^

3.

+

cfc1

a?

-^ + (^

=

+ x [log^-]2

3^++2/=^ ote2

rfa?

dx*

dx

4. "i

dx? 5.

'

sin

log^

+

sin q

loga?.

CHAPTER

ORTHOGONAL

XVII.

TEAJECTOEIES.

MISCELLANEOUS

EQUATIONS. ORTHOGONAL

TRAJECTORY.

217. Cartesians. of a equationf(x,y, a) 0 is representative The to family of curves. problem we now propose is that of findingthe equation of another investigate of which each each member cuts family of curves in of the former family at rightangles. And member such a problem as this it has been alreadypointedout The

that

=

it is necessary

to

treat

that so familycollectively, a ought not to appear in

It has may

been

shown

be eliminated

all members

of the

first

the

constant particularizing the equation of the family. in Art. 17 1, that the quantitya between the equations

.

*dx

Let this eliminant

This

is the

'dy dx

be

differential

equation of

the

first

family.

DIFFERENTIAL

268

EQUATIONS.

x+yJL=a,

Here

cLx

.v2+y2 2x( x -\-y" ),

and, eliminating #,

Hence

the

new

=

differential

^2+ 2^-^2

or

be

equationmust

o,

=

(3)

...........................

ay

homogeneous equation, and the variables separableby the assumptiony vx. as However, this being the same equation (2) with that x and y are interchanged, must its integral which

is

a

become

=

another

set of

each circles,

of which

the

touches

#-axis

the

ception ex-

be

at

the

origin. Ex.

2.

Find

the

orthogonaltrajectoryof

the

curves

2

i

A

being the parameter

and

A must

(2)gives

so

that

of the

be eliminated

n\ --

family.

these

between

two

equations.

x(b*+ A)+yyl(a?+ A) 0, =

a2 + A

=

and Thus

the differential

equationof

the

family is

(at-b^yy or

x*-y*+xyyi-

=a2-52 ................

(3)

ORTHOGONAL

Hence

changing y^

TRAJECTORY.

into

differential

the

"

269

of the

equation

,

#1 familv

of

is trajectories 2

But

this

being

primitive,viz.

the

same

the

have

equation (3) must

as

(4) same

:

^

a

n

y2

_-.

i

~

*

i.e.a set of conic sections confocal Find

Ex. 3. cardioides

the

r=a(l"

the former

orthogonal trajectoriesof

0) for

cos

different

set.

the

of

values

family

of

a.

^

Here

r^

and, eliminating a, for the

Hence

with

l~ =

=

dr

0

sm

2

we familyof orthogonaltrajectories

have

must

"

1 dr

n

log

r

or

"

2

log

cos

+

"

constant,

2t

r=b(l+cosO),

or

family of oppositedirection.

another

coaxial

cardioides

whose

point in

cusps

the

EXAMPLES. the

1. Find

of the family of parabolas orthogonaltrajectories

4cM? for different values

#2

=

2.

Show

similar

that

for different values of ellipses-04-^,=m2

3. Find =

a.

orthogonal trajectoriesof

the a2

r

of

the

the

parabolas "

=1

of orthogonal trajectories

the

is s?

of

=Ayb

.

equiangular spirals

a.

of the orthogonal trajectories -f cos

m

family

b2

ae^cota' for different values of 4. Find

the

9 for different values

confocal and of

a.

coaxial

DIFFERENTIAL

270 5. Show

that

EQUATIONS.

the families

of

curves

orthogonal.

are

6. Show r

sin2a

that

the

curves

a(cos 0

=

a)

cos

"

and

r

sinh2/? a(coshft =

"

cos

0)

orthogonal.

are

7. Show

form

that

if f(x+iy)

that

for any cosh

at

+

u

iv

the

curves

orthogonalsystems.

8. Prove

cut

=

x

family /z coth rightangles. the

SOME

220.

The

x

cosec

"

y

"

cosech

of /z the

value

constant

p cot y x

cos

y

=

constant

=

constant

family of

[LONDON, 1890.]

IMPORTANT

DYNAMICAL

EQUATIONS.

equation

general form of the equation of motion particleunder the action of a central force. is the

Multiplying by

2-^and

integratingwe

dO

which

and

we

may

the solution

221.

curves

write

as

is therefore

Equations

effected.

of the form

have

of

a

SPECIAL

SOME

have

alreadybeen

FORMS.

discussed

as

271

beinglinear

with

stant con-

coefficients. The

solution may

Multiplyby

sin

however

n9, which

be conducted

will be found

thus

to be

: "

integrating

an

factor.

Integrating, sin

nO^. d6 -

n(" +

nO=

sin n"dff f*f(ff)

+ A.

Jo

nO is cos Similarly, first integral is cos

cos

nu

factor integrating

an

sin nO=

nu

d\j

and

the

ing correspond-

f'f(ff) nO'dO'+B. cos

J o

Eliminating-^L du

nu

ef(0') sin n(0

=

-

O')d0f+ Bsmn6-A

cos

n6.

0

222. The mass

equationof

often takes

motion

such

some

form

of

a

body

of

changing

as

d! dt ^

and

for this

will equation"f"(x)-rr

be found

to be

factor. integrating

leads at

once

j^x)"l"(x)dxA, i{""(^' J2=

to

+

1

^Xto J

and

the variables

are

separated.

an

DIFFERENTIAL

272

FURTHER 223.

EQUATIONS.

ILLUSTRATIVE

1.

by reducingto alreadydiscussed by

forms

or

Ex.

be solved

Many equationsmay

other of the known artifices. special

one

EXAMPLES.

^=f(ax+by). '

ax

Let

ax+by=z.

Then

=

dx

dx

Thus dx

dz

d*..

x+A=l

or

Ex.2. dx\

y+a

J

+

a

bf(z)

dx

Put

xy=z.

rpi

dy

dz

dx

dx

y+^^=-y-,

Then

.

dz Z

1

, X-j- +-5-

=

dz

dx

or

dx which

is of Clairaut's

form, and

the

completeprimitiveis .

Ex.3.

Solve

e^

-

\dx

dx) Let

Then,

6^ =

since this

equation may dx

77,

he

ex

=

s-

arrangedas \e*dx

ILL

we

write it as

may

which written

USTRA

being of

UTIONS.

SOL

Tl VE

273

?7

Clairaut's form

the

completeprimitivemay

or

Ex.

4.

--

(an. equationoccurringin

Then

and

,v=*Js

Put the

Solid

y

Geometry).

"Jt.

=

equationbecomes

ds

giving

,

t=

ds as

which

is of Clairaut's form

and t-sG

has the

completeprimitive

BC ~

1+2(7'

and

singularsolution

the four

lines straight

9"J-Jy Ex. 5,

Solve the

equation dx

E. I. C.

S

be

DIFFERENTIAL

274 Let

then

the transformation

is known

x

by

EQUATIONS.

be such

direct

that

as integration

a

function

of

t.

dy

d^dL_

Now

dx

*

'

and dx*

f (^ax^yJ^axd4 }dx* dt* dx dt

Thus and

.

the

givenequationthus

reduces

whose

cos

and

of

is

solution is y=A sin qt + B when the value of t in terms

complete. we [Ifa be positive

,

to

"^, x

is

the substituted,

solution

have dx

1

,. "

"

-[=.

""~j= sinh^Wa) If

a

be

negativewe 1

=

t.

have dx

,=dt"

V-a

are

differential Solve the simultaneous Ex. 6. linear with constant coefficients)

equations(which

DIFFERENTIAL

276

EQUATIONS.

whence

operatingupon these in turn by eliminate y and obtain we subtracting, [(D2+ 16)(D2+ 9)+

15

+ 40Z)2 + (Z"4

or

by

3D

and

Z)2" 0, =

1 44"

0,

=

+ 36)^=0, (D*+ 4)(D'2

i.e. whence

x

=

A sin

the Differentiating the second

~2t +

B

2" + C sin 6" + D

cos

whence

we

cos

6*.

three times first equation and subtracting have differential coefficients of y, we

to eliminate

*"

dt

viz.

D2 + 9 and

obtain

the

value

without

of y

any

new

constants,

:

"

y=-%B

sin 2t + 2 A

2t + i"D

cos

sin 6"

-

^-

EXAMPLES. Solve i.

the

equations

2. 2^-(i-*)y"=**. 'cte

3.

4.

5.

(1-

.

2 8. Obtain

the

2

integralsof

cosy

following differential

the

tions equa-

: "

+

9y

-

25

cos

^

[I.C. S, 1804.]

EXAMPLES.

9.

simultaneous

the

Integrate

277

system

4=0. _

10.

Find

of

inclination to

the 11.

as

the

12.

radius

the of

product Find cube

Show of

the of

the

the

curve

current

the

of

the

cosine

in

of

the on

in

which

to

the

tangent

coordinates

form

that curvature

of

form

the

of for

the

inclination

is

7/-axis

of

is

the

y

oc

log

sec ~.

the

proportional

of

constant

the

curvature

the

tangent

projection

length

(l)Soclogtan(?+|), (2)

of

point.

which

for

tangent

#-axis

which

curve

curve

the

the

the

varies to

the

of

the

ANSWEES.

I.

CHAPTER

PAGE

1.

Area

=

e6-ea.

12.

Area=ia2tan

3.

0,

4.

Vol.^. 5

2.

Vol.

Vol.

=-(e*b-e2n).

6.

-a3tan2"9.

=

5.

Vol.=f7ra3.

(a) Vol.

=|

Vol.

=

1 TT

VoL

(8)

*

1 =

-

JL25

Vol.

u

=JL t)

7,

"7Tfia3.

8.

Mass

of

half

the

spheroid

=

J?r/xa262.

279

ANSWERS.

II.

CHAPTER PAGE

17.

a

2.

^Y.

6. 1.

3.

?^1.

7.

x/2-1.

4.

Ioge|.

8.

|. PAGE

"2 X

"'

*'

_c.o

ft C"

r!000

r!00

Ht;

X

+

23.

r!001

"

loo' Tooo' Tool'

_^

_^ 100'

10'

10.

98

PAGE

25.

PAGE

26.

"

2J, 2.

a

logx,

~j

a

log^ +#,

2. 3.

logtan"1^,logsin'1^,log(log^).

(sin6 -sin a).

CALCULUS.

INTEGRAL

280

PAGE

9

4+

"

log2J

aT

log3'

4

+I

log 6

logo2' log tan

4.

_?

""

5.

+

28.

^,

log sin

^

-

cosec

sin-1*,1 tan-,

l86^! an-^,

7. 8.

-log^

+

e*), log(logsin x\

III.

CHAPTER PAGE

32.

1.

sine*, sin#n, sin(log^).

3.

asin^+-tan-1^4,

cose*

-a

+6

log cosh

#.

4 4.

J_ tan-x-JL-. V2

2

_-,

6.

^2

L

.

PAGE /-

2.

cosh^+l),

-1?,

si

sin-V^-

3

"

-

8.

~.

-

"

V*

41. -

g

-H-.-fJain-1*,

'

v

^.

ANSWERS.

3.

-Vl=F, x/^,

281

s

4.

i(^2+1)4,

6.

-^2,| sinh-1^ I siii-1^-2\/r^-iWl

+

2\/l+

#2 +

7.

xyJ\^3?,4 cosh-1-

+

^"?x/5?^

2

8.

^logtan^,

2

-logtana"r+", ^logtan(-+.A 2

CL

\4

/ \

f

13.

),J logtan

x

-

Iog[log{log(loga7)}], log(log^), log{log(log,^)},

CHAPTER

IV. 47.

PAGE

^7

sinh

x

cosh #,

"

x (2+ #2)sinh

x

2"r

sm

"

Zx cosh

x.

2^

CQS

,

in 3a? + 9 sin 3.

J(sin2a?

-

2^

cos

+ cos a?)

3^ + 27

cos^].

2^), sin 4.f

sin 6^7

~""

~""^~~

/cos 2^7 _

cos

4^

cos _

~~

~~

6^7

5.

^sin^

--^sin^-tan-^), 2

6.

CALCULUS.

INTEGRAL

282

v

^2(a2

^2)~sin^-tan-1-,

+

q + r-p,

r+p-q,

sin-1*?+

x/f^;2

p + q-r, 7T2 7T,

of n,

for the values ct /

\

I.

tan-a4).

-

5

-p-q-r.

2

A 7T2-4.

_,

9.

(1 ^2). -

-

339

PAGE

51.

3(^2+ 2)coshx, 5(^4+ 1 2^72+ 24)sinhx. -

(rf+

20^"3+ 1 20#)cosh x

-

6)sinx, _

84\V

\ 2

2/

J{2(2^3 3^)sin2^ -

1.

(a) (m2+ 4

-

+

52.

cos^-sn-g,

where

3

\

3

\

tan

^

log cos

+

(d) ^tan"1^ (a) x

"

"

/

(/)

Jlog(l+^2).

Si

=

^74

1

tan

-"

x

.tT3 3^7 "

,

"

sec"1^

-

_i

lx

"

"

"

cosli"1^.

\/l" ^sin"1^.

+ cos0)-sin"9-logtan (b) 6"(sec(9

(c) 2

where cos (c?) (sin"/" "^" "/"), "

"

\

(e)

x.

A-

/

"

/

(c)^

3)cos

where #=sin#. l)~^rnecos(^-cot-1m),

gsmg

(6)

-

240, 265e-720.

5?r4+ 607T2

PAGE

6^2 +

(2^4

-

-

4

^=

+

Y

where

^

=

sin(9.

CALCULUS.

INTEGRAL

284

CHAPTEK

V.

PAGE

2.

"

.

+ 4# \ log(^'2

3.

+ 2). 5) taii"1^

+

-

4.

-log(3-.r).

5.

#-2log(#2 +

6. 2#

-

2.r +

2)+ 3tan-1(."

+ 6^7 + 10)+ f log(#2

PAGE

^ (iii.)

58.

11

tan-1^ H- 3).

62.

-

_

ct

x (ix.)

"

o

giZ^

7 a^T

+

log(^ q.)+etc. -

-

-

e 1

27

(^Tiy

+

+ 8

(^-l)2 8(^-l)

l0a^"^+ I_L.-?-JL_. g^-l+

16

j B

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