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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
n
n
x
x dx C
n
∫
dx lnx C
x
∫
c dx = cx +C
∫
2
x
xdx = +C
∫
3
2
x
x dx = +C
∫
2
dx C
x x
∫
x x
xdx = +C
∫
2
arctan
dx x C
x
∫
2
arcsin
dx x C
x
∫
Integrals of Trigonometric Functions
sin x dx = − cosx +C
∫
cos x dx = sinx +C
∫
tan x dx = ln secx +C
∫
sec x dx = ln tan x + secx +C
∫
( )
2
sin sin cos
x dx = x − x x +C
∫
( )
2
cos sin cos
x dx = x + x x +C
∫
2
tan x dx = tanx − x +C
∫
2
sec x dx = tanx +C
∫
Integrals of Exponential and Logarithmic Functions
ln x dx = x lnx − x +C
∫
( )
1 1
2
ln ln
n n
n
x x
x x dx x C
n
n
∫
x x
e dx = e +C
∫
ln
x
x
b
b dx C
b
∫
sinh x dx = coshx +C
∫
cosh x dx = sinhx +C
∫
2. Integrals of Rational Functions
Integrals involving ax + b
( )
( )
( )
( )
1
n
n
ax b
ax b dx
a
fo n
n
r
∫
dx lnax b
ax b a
∫
( )
( )
( )( )
( ) ( )
1
2
n n
a n x b
x ax b dx ax b
a n n
for n n
∫
2
ln
x x b
dx ax b
ax b a a
∫
( )
( )
2 2 2
ln
x b
dx ax b
a ax b a
ax b
∫
( )
( )
( )( )( )
( )
1 2
n n
a n x b
x
dx
ax b a n n
for n
ax b
n
−
∫
( )
( )
2
2
2
3
2 ln
ax b
x
dx b ax b b ax b
ax b a
∫
( )
2 2
2 3
2 ln
x b
dx ax b b ax b
ax b a
ax b
∫
( ) ( )
2 2
3 3 2
ln
x b b
dx ax b
ax b a
ax b ax b
∫
( )
( ) ( ) ( )
( )
3 2 1
2 2
3
n n n
n
ax b b a b b ax b x
dx
n n
fo
n
a
r n
a
x b
− − −
∫
( )
ln
ax b
dx
x ax b b x
∫
( )
2 2
ln
a ax b
dx
bx x x ax b b
∫
( )
( )
2 2 2 3
2
ln
ax b
dx a
x b a xb ab x b
x ax b
∫
Integrals involving ax
2
+ bx + c
2 2
1 1 x
dx arctg
a a x a
∫
2 2
ln
ln
a x
for x a
a a x
dx
x a x a
for x a
a x a
∫
4. Integrals of Logarithmic Functions
ln cxdx = x lncx −x
ln( ) ln( ) ln( )
b
ax b dx x ax b x ax b
a
2 2
ln x dx = x ln x − 2 x ln x + 2 x
1
ln ln ln
n n n
cx dx x cx n cx dx
−
2
ln
ln ln ln
ln!
i
n
x dx
x x
x i i
∞
=
1 1
ln 1 ln ln
n n n
for n
dx x dx
n
x n x x
− −
1
2
ln 1
n
l 1
m m
x
x xdx x
m
m
for m
1
1
ln
ln
ln 1
n
m
n n m m
x x n
x x dx x x dx
m
r
m
fo m
−
1
ln ln
n n
x x
dx for n
x n
2
ln
ln
n
n x
x
dx for n
x n
1 2 1
ln ln 1
m m m
x x
dx
x m x m
for
x
m
− −
1
1
ln ln n
l
n n n
m m m
x x x
n
dx dx
m x m x x
for m
−
−
ln ln
ln
dx
x
x x
1
1 ln
ln ln 1
ln
i i
i
n
i
n x
dx
x
i i x x
∞
=
1
ln 1 ln
n n
dx
x x n
f
x
or n
−
2 2 2 2 1
ln ln 2 2 tan
x
x a dx x x a x a
a
−
sin ln sin ln cos ln
x
x dx = x − x
cos ln sin ln cos ln
x
x dx = x + x
5. Integrals of Trig. Functions
sin xdx = −cosx
∫
cos xdx = −sinx
∫
2
sin sin 2
x
xdx = − x
∫
2
cos sin 2
x
xdx = + x
∫
3 3
sin cos cos
xdx = x − x
∫
3 3
cos sin sin
xdx = x − x
∫
ln tan
sin 2
dx x
xdx
x
∫
ln tan
cos 2 4
dx x
xdx
x
∫
2
cot
sin
dx
xdx x
x
∫
2
tan
cos
dx
xdx x
x
∫
3 2
cos 1
ln tan
sin 2sin 2 2
dx x x
x x
∫
3 2
sin 1
ln tan
cos 2 cos 2 2 4
dx x x
x x
∫
sin cos cos 2
x xdx = − x
∫
2 3
sin cos sin
x xdx = x
∫
2 3
sin cos cos
x xdx = − x
∫
2 2
sin cos sin 4
x
x xdx = − x
∫
tan xdx = −ln cosx
∫
2
sin 1
cos cos
x
dx
x x
∫
2
sin
ln tan sin
cos 2 4
x x
dx x
x
∫
2
tan xdx = tanx −x
∫
cot xdx =ln sinx
∫
2
cos 1
sin sin
x
dx
x x
∫
2
cos
ln tan cos
sin 2
x x
dx x
x
∫
2
cot xdx = − cotx −x
∫
ln tan
sin cos
dx
x
x x
∫
2
ln tan
sin 2 4 sin cos
dx x
x x x
∫
2
ln tan
sin cos cos 2
dx x
x x x
∫
2 2
tan cot
sin cos
dx
x x
x x
∫
( )
( )
( )
( )
2 2
sin sin
sin sin
m n x m n x
mx nxdx
n m n
m n
m
∫
( )
( )
( )
( )
2 2
cos cos
sin cos
m n x m n x
mx nxdx
n m n
m n
m
∫
( )
( )
( )
( )
2 2
sin sin
cos cos
m n x m n x
mx nxdx
m n m n
m n
∫
1
cos
sin cos
n
n
x
x xdx
n
∫
1
sin
sin cos
n
n
x
x xdx
n
∫
2
arcsin xdx = x arcsin x + 1 −x
∫
2
arccos xdx = x arccos x − 1 −x
∫
( )
2
arctan arctan ln 1
xdx = x x − x +
∫
( )
2
arc cot arc cot ln 1
xdx = x x + x +
∫
