Definite and Indefinite Integration

KEY 1.

CONCEPTS

DE FI NI TI ON :

If f & g are functions are functions of  of x such that g'(x) = f(x) = f(x) then  then the function func tion g is called called a PRIMITIVE ANTIDERIVATIVE ANTIDERIVATIVE O R INTEGRAL of f(x) w.r.t. x and is is writte wri tten n symbolically as

OR

f ci I  f(x) dx f(x) dx = g(x) + c <  c  — (g(x) + c) = f(x), = f(x), where  where c is called the  constant of integration, dx 2.

STANDARD RESULTS : \ nn+ l

r

(ax+b) i) J (ax + b)°dx= , \   a(n+l)

+c

j

(ii)

n*-l

=-

ax+b

a

/n (ax + b) + c  px+ q

iii) f  e

ax+b

dx = - e a

ax+b

r 1 n x+ (iv) I aP i dx = —  (a > 0) + c . p fn a

 + c

v) f si n(a x+b ) dx = -— co s( ax +b ) + c a

(vi) J  c o s ( a x + b )dx= - sin(ax +b ) + c a

vii) j tan(ax + b) dx = —  In sec (ax + b) + c a

(viii) f   cot(ax+ b) dx = - /n sin(ax+ sin(a x+ b)+  c a

2

2

ix) J sec  (ax + b) dx = —  tan(ax + b) + c a

(x)  j cosec (ax  + b) dx = _  1  1  cot(ax + b)+ c

xi) J sec (ax + b) . tan (ax + b) dx = —   sec (ax + b) + c a xii) J  cosec co sec (ax + b) . cot co t (ax + b) dx =

cosec (ax + b) + c a In tan {

+yj

+ c

xiii) J secx dx =I n (secx + tanx) + c

OR 

xiv) J cosec x dx = In In (cosecx (cose cx - cotx) + c

OR  In tan ^ + c OR - In (cosecx (cose cx + cotx) 2

xv) J sinh sinh x dx = cosh x + c (xvi) J cosh x dx = sinh sinh x + c (xvii) J  sech x dx = tanh x + c 2

xviii) J  cosech x dx = - coth x + c

(xix) J sech x . tanh x dx dx = - sech x + c

xx) J cosechx. cosechx. cot hxd x= -co se chx + c 

(xxi) j"

xxii) J

xxiv) JJ ' xxv) J

xxvi) |

dx

„ 2„ = - ta n a +x 2

-1

— + — + c

72 =ln =l n + 2 L x  + a  +  a

dx

dx a2 -x 2

1 I , dX > Jf Vax  + bx + c j^/x+1 r~T 1 ((x2 +3x+3

^ r Q.48

dx Y x3V(l + x)3

2

dx

Q.47 JJ  (x-a)V(x-a)(x-(3) 0) V 7 '

 Definite & Indefinite Integration

 [18]

Q.23

(2x+ 3 )

3/2 Q.24 J |x. sinTtxl dx

x

Evaluate: f  f  d x . I (1 + cos x)  p+ qn

Q.25 Show that J | cos x| dx = 2q + sinp where q s N  & -— < p b>0

 Definite & Indefinite Integration

 [18]

 A

Q.45

Let  f(x) = - j /n cos y dy then prove that  f(x) = 2f

71 v4

2,

-2f

X

4  _  2

-x/n2.

Jt/2

Hence evaluate J

sec

 t dt

~ .^ , r x. lnx , , r a x. dx Q.46 Showthat J f ( - + - ) . dx = ln a. Jf (- +- ). x a x x a x 1

Q. 47

-(2x332 +x 99 8 +4x 1668-sinx 69 1 ) dx Evaluate the definite integral, j 666 1 + x -l

Q.48

Prove that

0» IM

(a) } V(x -a )( P- x)

 p (c)5{

 p

dx xA/(x-a)()3-x)

V^P

4 cos x Q.49

If f(x) = (cosx-1)

Q.50 Evaluate:

/ntan_ - 1 xx

Je/ n t a n

1

1 2

2

2 (cosx + 1)   (cosx-1) , find Jf( x) dx

(cosx + 1)2 1

x.dx

(cosx + 1)2

cos 2 x

-sin _1 (cosx)dx. -1,

o

 EXERCISE-III Q.l

If the derivative of f(x) wrtx is — - then show that f(x)is a periodic function.

Q.2

Find the range of the function,  f(x)= f 

Q.3

A function  f is defined in [-1,1] as f (x) =  2 x sin —   - cos — ; x ^ 0 ; f(0) = 0 ;

f(x)

Sm X

—7 .

1 —   2t cosx + t

X

X

f (I/71) — 0 .  Discuss the continuity and derivability of f at x=0. Q.4

Let f(x) = [

-1 if - 2 < x < 0 I and g(x) = J f(t) dt. Define g (x) as a function ofx and test the x -* 1 if 0 < x < 2 _2

continuity and differentiability of g(x) in (-2,2). Q.5

Prove the inequalities : (a) 0 < J 0

x 7 dx_

Ml

/3

1

(b) 2 e~1/4 < } ex2"x dx < 2e2.

8

2s (c) a< f — < b then find a & b. i 10+3cos x Q.6

dx

(d) ^ < J 2 + x2 " 6 —2 0

Determine a positive integer n 2 prove that

- n (n - 1) Un _ 2 - 2 n(2n - l)U n _  p

1 further  if Vn - J e x . U n dx, prove that when n > 2, Vn + 2n(2n- l). V n _ r  n(n- 1) Vn _ 2 = 0 0 f J?nt Ait  Q. 16 If  J —2—2 .2 0 X ~F~ T equation.

=

%£n22 71 4

"1-x

Q.17 Let f(x)=

if

( x > 0) then show that there can be two integral values of 'x' satisfying this

0

(D) [g(x)/g(7l)]

( Q - 1 + V2

(D )l + V2

:

(B) - 1 + VJ

e" The value of J ————— dx is

(c)

I

sinx

H e Let — F(x) = dx x values ofki s

(d)

Q.3

4

? , x > 0 . If f — ' x .

dx = F (k) - F (1) then one ofthe possible

2x

0 + s i n x ) d x 1 + COS X

(e)

Determine the value of J „

(a)

If ff(t) dt = x + f t f ( t ) dt , then the value of /( l) is

[JEE '9 7, 2 + 2 + 2 + 2 + 5]

(A)° 1/ 2

(B) 0 (C) 1 (D) -1/2 i f  Y  \ i Prove that f tan"1 dx = 2 f tan 1 x dx . Hence or otherwise, evaluate the integral 2 i U-x+x ; i i Jtan _1 (l-x+x 2 )dx [JEE'98,2 + 8]

(b)

Q

4

E

r

a

l

u

a

t

e

(ilZ?a«sa/  

[

Classes

 Definite & Indefinite Integration

R

E

E

'

9

8

-

6

1

 [18]

Q.5

(a)

If for al real number  y, [y] is the greatest integer less than or equal to y, then the value of the 371/2 integral j [2 sinx] dx is: (A) - n

(B) 0

(C)

(D) *

2

2

3,1/4

rlY f — — — is equal to : L 1+ cosx

(b)

(A) 2 /N (c)

(C)

(B) - 2

x3 + 3x + 2 , Integrate: J — dx (x 2 +l ) (x + 1)

T

R 

ii

(d)

Integrate: J

„cosx .

e

+e

-cos x

[JEE '99, 2 + 2 + 7 + 3 (out of200)]

-dx

71/6

Q.6\\

Evaluate the integral J

Q.7

(a)

V3cos2x cosx

The value of the integral (A) 3/2

[REE'99, 6] log e x

j

dx is:

(B) 5/2

(C) 3

(D) 5

 x 

(b)

r 1 ' 1 Let g( x) = J f (t ) dt, where f is such that - < f( t)