Integration Cheat Sheet

 

INDEFINITE INTEGRATION

  Forrmu Fo mula lass :

∫x dx 1 2.∫ dx x x 3.∫e dx x 4.∫a dx n

1.

=

xn + 1 n+1

=

logx

=

ex

ax = loga

5.

sinx dx = - cosx 6. cosx dx = sinx

∫ ∫ 7.∫tanx dx = log secx = - log cosx 8.∫cotx dx = log sinx = - log cosecx π x 9.∫secx dx = log secx + tanx = log tan( +  ) 4 2 10.

cosecx dx = log cosecx - cotx

∫ 11.∫se secx cxta tanx nx dx = secx 12.∫co cose secx cxco cotx tx dx = - cosecx 13.∫sec x dx = tanx 14.∫cosec x dx = - cotx 2

2

∫ √ 1 - x

15.

1

2

dx = sin - 1 x

=

(  )

log tan x2

 

∫ √ 1 - x dx = cos x  1 17.∫ dx = tan x 1+x -1 18.∫ dx = cot x 1+x 1 19.∫ dx = sec x √  - 1 x x- 1 20.∫ dx = cosec x x√ x - 1 -1

16.

-1

2

-1

2

-1

2

-1

2

-1

2

x dx = x

∫

21.

⇒

x

∫ex  (f(x) + f' (x))dx

ex f(x)

=

Spec Sp eciial In Intteg egra rals ls :

∫ √ x + a 1

1.

2

∫ √ x - a

2

1

2.

2

2

∫ √ a - x 1

3.

2

2

dx = log x +√ x2 + a2 dx = log x +√ x2 - a2 dx = sin - 1

1

4.

∫ 1 5.∫ x -a

2

∫ a -x

6.

1

2

2

∫√ x

7.

2

∫√ x

8.

2

(  )

1

x2 + a2 dx = 2

x a

dx = dx =

+a

2

-a

2

a tan 1 2a 1 2a

dx =

dx =

-1

x

x+a a - xa x-

√ x 2

x

(  )

x-a x+a

log

log

x a

2

√ x 2

2

+a

2

-a

2

+

-

a2 2

a2 2

log x +√ x2 + a2

log x +√ x2 - a2

 

 

∫√ a

2

9.

2

-x

x

√ a 2

dx =

2

2

-x

+

a2 2

sin - 1

x a

(  )

Stan St anda dard rd Su Subs bsti titu tuti tion onss : →For the terms of the form : 2

2

2

2

(i) x + a   (or) √ x + a

put x = atanθ   (or) x = acotθ

(ii) x2 - a2  (or) √ x2 - a2 put x = asecθ   (or) x = acosecθ

(iii) a2 - x2  (or) √ a2 - x2 put x = asinθ  (or) x = acosθ (iv) If both √ a + x , √ a - x are present then put x = acosθ (v) For the type √  (  (x - a)(b - x) put x = acos2 θ + bsin2 θ

Some So me Us Usef efu ul Tri rig gnom omet etrric For orm mul ula as : 2

1. 1 + cos2θ = 2cos

θ 2 2. 1 - cos2θ = 2sin θ 3. cos3x = 4cos3 x - 3cosx 4. sin3x = 3sinx - 4sin3 x 5.

sin3x sinx

2

= 3 - 4sin

θ

Integration By Parts : If u and v are two functions of x then

∫

∫

uv dx = u v dx -

∫

d   (u) dx

∫v dx dx

Choice Of 1st and 2nd Functions : The 1st function is the function which comes first in the word 'ILATE' I - In Inve vers rsee tr trig igno nome metr tric ic fu func ncti tion onss

 

  L - Lo Logar garthm thmic ic fun funct ction ion A - Alg Algebr ebrai aicc fun funct ction ion T - Tr Trign ignome ometri tricc fun functi ction on E - Exponential ⇒

If one of the two function is not integrable directly then take

that function as 1st. ⇒ If onr of the function is not directly integrable then 'unity(1)' is taken as second function. Forrmu Fo mula lass :

∫(f(x) + xf' (x))dx

1.

=

ax

e sin(bx + c)dx =

2.

xf(x) eax

asin(bx + c) - bcos(bx + c)

a2 + b2 eax ax 3. e cos(bx + c)dx = 2 acos(bx + c) + bsin(bx + c) 2 a +b

∫ ∫

∫a sin(bx + c)dx

4.

x

∫a cos(bx + c)dx

5.

x

=

=

ax 2

2

(loga loga)) + b

loga.sin(bx + c) - bcos(bx + c)

ax loga.cos(bx + c) + bsin(bx + c) 2 2 (loga loga)) + b

Note : → If the integral is of the form :

∫ a + bsinx 1

dx   (or)

∫ a + bcosx 1

x

(  ) = t

Put tan

2

working rule : tan

(  ) = t x 2

Diff Di ffer eren enti tiat atin ing g w. w.r. r.to to x,

dx   (or)

∫ asinx + bcosx + c dx 1

 

1

2 se c 2

x

(  )dx sec2

(  ) x 2

2dt

dx =

2

1 + tan

( x ) 2

2dt

dx =

sinx =

dt 

2dt

dx =

⇒

=

2

1 + t2

( x ) x 1 + tan (  ) 2tan

2

2

2

⇒

sinx =

2t 1 + t2

( x ) x 1 + tan (  ) 2

1 - tan

cosx =

2

2

2

⇒

→

cosx =

∫

∫

→

1-t 1+t

2

2

acosx + bsinx dx = ccosx + dsinx

(

ac + bd x + c2 + d2

 )

acosx + bsinx + L dx = ccosx + dsinx + k

(

(

ac + bd x + 2 2 c +d

 )

ad - bc log lo g De Deno nomi mina nato torr c2 + d2

 )

(

∫ ccosx + dsinx + k

 )

1

+ (L - Ak) ⇒

ad - bc log lo g De Deno nomi mina nato torr 2 2 c +d

where A =

(

CBSE CBSE

ac + bd c2 + d2

 )

(or) Nr = A (Denominator) + B(Denominator)

#If the integral is of the form of

∫ acos x + bsin x + csinxcosx dx   (or) 1

2

2

∫ a + bsin x 1

2

dx   (or)

∫ a + bcos x dx   (or) 1

2

 

∫ acos x + bsin x + c dx   (or)  1

2

2

∫ (acosx +1 bsinx bsinx))

2

dx

working rule : ⇒

Divide both numerator and denominator by cos2 x and then tak akee 't 'tan anx x = t' Intteg In egrrati tion on of im impr pro ope perr fr fra act ctiion onss : ⇒

f(x) is said to be improper when degree of f(x) > = degree of g g(x)

(x) ⇒

Pro roccee eed d thr hro ough di div vidi din ng nume merrato torr by de den nom omin ina ato torr Dividend = Divisor × quotient + remainder

#If the integral is of the form : +q px + q dx   (or) ∫ ∫ axpx+ bx +c √ ax + bx + c 2

2

(or)

∫(px + q)√ ax

working rule : d Numerator = A. dx (Denominator) + B ⇒

where A and B are constants to be determined

Intteg In egrrati tion on of Ir Irrrati tion ona al Al Alg geb ebrraic fr fra acti tion on : → Irration functions of the form : 1

(ax + b) n can be evaluated by the substitution ⇒

tn = ax + b

2

+ bx + c

 

→ For

∫ (x - k) √ ax + bx + c 1

r

1

⇒ (x - k) =

→ For ⇒

x2 =

t

∫ (ax + b)√ cx + d 1

2

put x =

dx su subs bsti titu tute te  

2

dx

2

1

⇒

t

-1

dx =

2

t

dt

1

t2

#Integration of the type :

∫sinm x.cosn x dx (i) If one of them is odd then substitute for the even power (ii) If both are odd substitute either of term (iii) If both are even use trignometric identities only

Red edu ucti tion on For orm mul ula ae : → In =

∫xn eax dx then ax

⇒

a

→ In = ⇒

(a

In = e xn - n .In - 1

∫sinn xdx

In =

→ In =

)

- sin

n-1

x.cosx

n

+

n-1 .In - 2 n

∫

cosnxdx

cosn - 1x.sinx n-1 ⇒ In = + .In - 2 n n

 

→ In =

∫

tann dx 

tann - 1 x ⇒ In = - In - 2 n-1

→ In =

∫cotn dx -1

⇒

In =

→ In =

- cotn

x - In - 2

n-1

∫secn xdx

secn - 2 x.tanx n-2 ⇒ In = + .I n-1 n - 1 n - 2 

→ In = ⇒

In

cosecn xdx

∫ - cosec =

n-2

n-1

x.cotx

+

n - 32 .I n - 1 n - 2 

Note : → Don't forget to add integration constant 'c' for all integration formulas. - - - - - - - - - - - Prepared

by Vijay - - - - - - - - - - -