Integration Formulas

1. Common Integrals

Indefinite Integral

Method of substitution

( ( )) ( ) ( )f g x g x dx f u du′ =∫ ∫

Integration by parts

( ) ( ) ( ) ( ) ( ) ( )f x g x dx f x g x g x f x dx′ ′= −∫ ∫

Integrals of Rational and Irrational Functions

1

1

nn x

x dx Cn

+

= ++∫

1lndx x C

x= +∫

c dx cx C= +∫

2

2

xxdx C= +∫

32

3

xx dx C= +∫

2

1 1dx C

x x= − +∫

2

3

x xxdx C= +∫

2

1arctan

1dx x C

x= +

+∫

2

1arcsin

1dx x C

x= +

−∫

Integrals of Trigonometric Functions

sin cosx dx x C= − +∫

cos sinx dx x C= +∫

tan ln secx dx x C= +∫

sec ln tan secx dx x x C= + +∫

( )2 1sin sin cos

2x dx x x x C= − +∫

( )2 1cos sin cos

2x dx x x x C= + +∫

2tan tanx dx x x C= − +∫

2sec tanx dx x C= +∫

Integrals of Exponential and Logarithmic Functions

ln lnx dx x x x C= − +∫

( )

1 1

2ln ln

1 1

n nn x x

x x dx x Cn n

+ +

= − ++ +

∫

x xe dx e C= +∫

ln

xx b

b dx Cb

= +∫

sinh coshx dx x C= +∫

cosh sinhx dx x C= +∫

2. Integrals of Rational Functions

Integrals involving ax + b

( )( )

( )( )

1

11

nn ax b

ax b dxa

fo nn

r

++

+ =+

≠ −∫

1 1lndx ax b

ax b a= +

+∫

( )( )

( )( )( ) ( )

1

2

11

2,

12

n na n x bx ax b dx ax b

a n nfor n n

+≠ −

+ −+ = +

+ +≠ −∫

2ln

x x bdx ax b

ax b a a= − +

+∫

( ) ( )2 2 2

1ln

x bdx ax b

a ax b aax b= + +

++∫

( )

( )

( )( )( )( )

12

12

1,

21

n n

a n x bxdx

ax b a n nfor n

ax bn

−≠

− −=

+−

+ − −≠ −∫

( )( )

222

3

12 ln

2

ax bxdx b ax b b ax b

ax b a

+ = − + + + +

∫

( )

2 2

2 3

12 ln

x bdx ax b b ax b

ax baax b

= + − + − ++

∫

( ) ( )

2 2

3 3 2

1 2ln

2

x b bdx ax b

ax baax b ax b

= + + − ++ +

∫

( )

( ) ( ) ( )( )

3 2 122

3

21

3 2 11, 2,3

n n n

n

ax b b a b b ax bxdx

n nfo

nar n

ax b

− − − + + + = − + − − − −+

≠∫

( )1 1

lnax b

dxx ax b b x

+= −

+∫

( )2 2

1 1ln

a ax bdx

bx xx ax b b

+= − +

+∫

( ) ( )2 2 2 32

1 1 1 2ln

ax bdx a

xb a xb ab x bx ax b

+= − + −

++ ∫

Integrals involving ax2 + bx + c

2 2

1 1 xdx arctg

a ax a=

+∫

2 2

1ln

1 2

1ln

2

a xfor x a

a a xdx

x ax afor x a

a x a

−< +

= −− >

+

∫

2

2 2

22

2 2 2

2

2 2arctan 4 0

4 4

1 2 2 4ln 4 0

4 2 4

24 0

2

ax bfor ac b

ac b ac b

ax b b acdx for ac b

ax bx c b ac ax b b ac

for ac bax b

+− >

− −

+ − −= − <

+ + − + + −− − = +

∫

2

2 2

1ln

2 2

x b dxdx ax bx c

a aax bx c ax bx c= + + −

+ + + +∫ ∫

( )

2 2

2 2

2 2

2 2 2

2 2

2 2ln arctan 4 0

2 4 4

2 2ln arctanh 4 0

2 4 4

2ln 4 0

2 2

m an bm ax bax bx c for ac b

a a ac b ac b

mx n m an bm ax bdx ax bx c for ac b

aax bx c a b ac b ac

m an bmax bx c for ac b

a a ax b

− ++ + + − >

− −+ − +

= + + + − <+ + − −

− + + − − =

+

∫

( ) ( )( )( )

( )

( )( ) ( )1 122 2 2 2

2 3 21 2 1

1 41 4n n n

n aax bdx dx

n ac bax bx c n ac b ax bx c ax bx c− −

−+= +

− −+ + − − + + + +∫ ∫

( )

2

2 22

1 1 1ln

2 2

x bdx dx

c cax bx c ax bx cx ax bx c= −

+ + + ++ +∫ ∫

3. Integrals of Exponential Functions

( )2

1cx

cx exe dx cx

c= −∫

22

2 3

2 2cx cx x xx e dx e

c c c

= − +

∫

11n cx n cx n cxnx e dx x e x e dx

c c

−= −∫ ∫

( )

1

ln!

icx

i

cxedx x

x i i

∞

=

= +⋅

∑∫

( )1

ln lncx cx

ie xdx e x E cxc

= +∫

( )2 2

sin sin coscx

cx ee bxdx c bx b bx

c b= −

+∫

( )2 2

cos cos sincx

cx ee bxdx c bx b bx

c b= +

+∫

( )( )1

2

2 2 2 2

1sinsin sin cos sin

cx ncx n cx n

n ne xe xdx c x n bx e dx

c n c n

−−

−= − +

+ +∫ ∫

4. Integrals of Logarithmic Functions

ln lncxdx x cx x= −∫

ln( ) ln( ) ln( )b

ax b dx x ax b x ax ba

+ = + − + +∫

( ) ( )2 2

ln ln 2 ln 2x dx x x x x x= − +∫

( ) ( ) ( )1

ln ln lnn n n

cx dx x cx n cx dx−

= −∫ ∫

( )

2

lnln ln ln

ln !

i

n

xdxx x

x i i

∞

=

= + +⋅

∑∫

( ) ( )( ) ( )( )

1 11

1

1ln 1 ln lnn n n

for ndx x dx

nx n x x− −

= − +−−

≠∫ ∫

( )( )1

2

ln 1n

11l

1

m m xx xdx x

m mfor m

+ = − + +

≠∫

( )( )

( ) ( )1

1lnln

1 11ln

nmn nm m

x x nx x dx x x dx

mr

mfo m

+−

= − ≠+ +∫ ∫

( ) ( )( )

1ln ln

11

n nx x

dx for nx n

+

= ≠+∫

( )( )

2

lnln0

2

nn xx

dx for nx n

= ≠∫

( ) ( )( )

1 2 1

ln ln 1

1 11

m m m

x xdx

x m x mfor

xm

− −= − −

− −≠∫

( ) ( )

( )

( )( )

1

1

ln ln n1

l

11

n n n

m m m

x x xndx dx

mx m x xfor m

−

−= − +

−−≠∫ ∫

ln lnln

dxx

x x=∫

( )( ) ( )

1

1 lnln ln 1

!ln

i ii

ni

n xdxx

i ix x

∞

=

−= + −

⋅∑∫

( ) ( )( )( )

1

1

ln 1 ln1

n n

dx

x x nf

xor n

−= −

−≠∫

( ) ( )2 2 2 2 1ln ln 2 2 tan

xx a dx x x a x a

a

−+ = + − +∫

( ) ( ) ( )( )sin ln sin ln cos ln2

xx dx x x= −∫

( ) ( ) ( )( )cos ln sin ln cos ln2

xx dx x x= +∫

5. Integrals of Trig. Functions

sin cosxdx x= −∫

cos sinxdx x= −∫

2 1sin sin 2

2 4

xxdx x= −∫

2 1cos sin 2

2 4

xxdx x= +∫

3 31sin cos cos

3xdx x x= −∫

3 31cos sin sin

3xdx x x= −∫

ln tansin 2

dx xxdx

x=∫

ln tancos 2 4

dx xxdx

x

π = +

∫

2cot

sin

dxxdx x

x= −∫

2tan

cos

dxxdx x

x=∫

3 2

cos 1ln tan

sin 2sin 2 2

dx x x

x x= − +∫

3 2

sin 1ln tan

2 2 4cos 2cos

dx x x

x x

π = + +

∫

1sin cos cos 2

4x xdx x= −∫

2 31sin cos sin

3x xdx x=∫

2 31sin cos cos

3x xdx x= −∫

2 2 1sin cos sin 4

8 32

xx xdx x= −∫

tan ln cosxdx x= −∫

2

sin 1

coscos

xdx

xx=∫

2sin

ln tan sincos 2 4

x xdx x

x

π = + −

∫

2tan tanxdx x x= −∫

cot ln sinxdx x=∫

2

cos 1

sinsin

xdx

xx= −∫

2cos

ln tan cossin 2

x xdx x

x= +∫

2cot cotxdx x x= − −∫

ln tansin cos

dxx

x x=∫

2

1ln tan

sin 2 4sin cos

dx x

xx x

π = − + +

∫

2

1ln tan

cos 2sin cos

dx x

xx x= +∫

2 2tan cot

sin cos

dxx x

x x= −∫

( )

( )

( )

( )2 2

sin sinsin sin

2 2

m n x m n xmx nxdx

n m nm n

m

+ −− +

+ −≠=∫

( )

( )

( )

( )2 2

cos cossin cos

2 2

m n x m n xmx nxdx

n m nm n

m

+ −− −

+ −≠=∫

( )

( )

( )

( )2 2

sin sincos cos

2 2

m n x m n xmx nxdx

m n m nm n

+ −= +

+ −≠∫

1cos

sin cos1

nn x

x xdxn

+

= −+∫

1sin

sin cos1

nn x

x xdxn

+

=+∫

2arcsin arcsin 1xdx x x x= + −∫

2arccos arccos 1xdx x x x= − −∫

( )21arctan arctan ln 1

2xdx x x x= − +∫

( )21arccot arc cot ln 1

2xdx x x x= + +∫