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Integration Formulas
1
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mathportal
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Integrals of Rational Functions
Integrals involving ax + b

( ax + b )n + 1
∫ ( ax + b ) dx = a ( n + 1)
n

1

( for n ≠ −1)

1

∫ ax + b dx = a ln ax + b
∫ x ( ax + b )

n

a ( n + 1) x − b

dx =
a

x

x

2

( n + 1)( n + 2 )

( ax + b )n+1

( for n ≠ −1, n ≠ −2 )

b

∫ ax + b dx = a − a 2 ln ax + b
x

b

1

∫ ( ax + b )2 dx = a 2 ( ax + b ) + a 2 ln ax + b
a (1 − n ) x − b

x

∫ ( ax + b )n dx = a 2 ( n − 1)( n − 2)( ax + b )n−1

( for n ≠ −1, n ≠ −2 )

2

x2
1  ( ax + b )

dx = 3
− 2b ( ax + b ) + b 2 ln ax + b 
∫ ax + b

2
a 



x2

∫ ( ax + b )2
x2

∫ ( ax + b )3
x2

∫ ( ax + b ) n

1 
b2 
dx = 3  ax + b − 2b ln ax + b −

ax + b 
a 


dx =

1 
2b
b2
 ln ax + b +
−
ax + b 2 ( ax + b )2
a3 


dx =

3−n
2− n
1−n
2b ( a + b )
b2 ( ax + b )
1  ( ax + b )
−
+
−
n−3
n−2
n −1
a3 


1

1

∫ x ( ax + b ) dx = − b ln
1

ax + b
x

1

a

∫ x 2 ( ax + b ) dx = − bx + b2 ln
1

∫ x 2 ( ax + b )2

ax + b
x


1
1
2
ax + b
dx = − a  2
+ 2 − 3 ln
 b ( a + xb ) ab x b
x


Integrals involving ax2 + bx + c
1

1

x

∫ x 2 + a 2 dx = a arctg a

a−x
1
 2a ln a + x

∫ x2 − a 2 dx =  1 x − a
 ln
 2a x + a

1






for x < a
for x > a











( for n ≠ 1, 2,3)

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2
2ax + b

arctan

2
4ac − b 2
 4ac − b

1
2
2ax + b − b 2 − 4 ac

dx = 
ln
∫ ax 2 + bx + c
 b 2 − 4ac 2 ax + b + b 2 − 4ac

− 2
 2ax + b

x

1

∫ ax 2 + bx + c dx = 2a ln ax

2

+ bx + c −

for 4ac − b 2 > 0
for 4ac − b 2 < 0
for 4ac − b 2 = 0

b
dx
∫ ax 2 + bx + c
2a

m
2an − bm
2ax + b
2
arctan
for 4ac − b 2 > 0
 ln ax + bx + c +
2
2
2a
a 4ac − b
4ac − b

m
mx + n
2an − bm
2ax + b

2
2
∫ ax 2 + bx + c dx =  2a ln ax + bx + c + a b2 − 4ac arctanh b2 − 4ac for 4ac − b < 0

m
2an − bm
 ln ax 2 + bx + c −
for 4ac − b 2 = 0
a ( 2 ax + b )
 2a


∫

1

( ax

∫x

2

+ bx + c

)

n

1

( ax

2

+ bx + c

)

dx =

2ax + b

( n − 1) ( 4ac − b2 )( ax 2 + bx + c )

dx =

n−1

+

( 2 n − 3 ) 2a
1
dx
2 ∫
( n − 1) ( 4ac − b ) ( ax 2 + bx + c )n−1

1
x2
b
1
ln 2
− ∫ 2
dx
2c ax + bx + c 2c ax + bx + c

3
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mathportal
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Integrals of Logarithmic Functions

∫ ln cxdx = x ln cx − x
b

∫ ln(ax + b)dx = x ln(ax + b) − x + a ln(ax + b)
2

2

∫ ( ln x ) dx = x ( ln x ) − 2 x ln x + 2 x
n
n
n −1
∫ ( ln cx ) dx = x ( ln cx ) − n∫ ( ln cx ) dx
i

∞ ln x
( )
dx
= ln ln x + ln x + ∑
∫ ln x
n =2 i ⋅ i !

dx

∫ ( ln x )n

=−

x

( n − 1)( ln x )

n −1

+

1
dx
n − 1 ∫ ( ln x )n −1

 ln x
1
x m ln xdx = x m +1 
−
∫
 m + 1 ( m + 1) 2


∫ x ( ln x )
m

∫

( ln x )n
x

n

dx =

dx =

x m+1 ( ln x )

n

m +1

−

( ln x )n+1

)

( for m ≠ 1)

n
n −1
m
∫ x ( ln x ) dx
m +1

2

ln x n
ln x n
( for n ≠ 0 )
∫ x dx = 2n
ln x
ln x
1
∫ xm dx = − ( m − 1) xm−1 − ( m − 1)2 xm−1

∫

( ln x )n
xm

( for m ≠ 1)

( ln x )n
( ln x )n−1
n
dx = −
+
dx
( m − 1) x m−1 m − 1 ∫ x m

dx

∫ x ln x = ln ln x
∞

dx

∫ xn ln x = ln ln x + ∑ ( −1)
i =1
dx

∫ x ( ln x )n
∫ ln ( x

2

=−

i

( n − 1)i ( ln x )i
i ⋅ i!

1

( for n ≠ 1)

( n − 1)( ln x )n−1

)

(

)

+ a 2 dx = x ln x 2 + a 2 − 2 x + 2a tan −1
x

∫ sin ( ln x ) dx = 2 ( sin ( ln x ) − cos ( ln x ) )
x

( for m ≠ 1)

( for n ≠ 1)

n +1

(






( for n ≠ 1)

∫ cos ( ln x ) dx = 2 ( sin ( ln x ) + cos ( ln x ) )

x
a

( for m ≠ 1)

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5
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Functions

∫ sin xdx = − cos x
∫ cos xdx = − sin x

cos x

x 1
− sin 2 x
2 4
x 1
2
∫ cos xdx = 2 + 4 sin 2 x
1
3
3
∫ sin xdx = 3 cos x − cos x
1 3
3
∫ cos xdx = sin x − 3 sin x

∫ sin

2

xdx =

dx

cos 2 x
x
∫ sin x dx = ln tan 2 + cos x

∫ cot

2

xdx = − cot x − x

dx

∫ sin x cos x = ln tan x
dx

x

1

π

∫ sin 2 x cos x = − sin x + ln tan  2 + 4 


dx

1

x

x

∫ sin x cos2 x = cos x + ln tan 2

x

∫ sin 2 x cos2 x = tan x − cot x

∫ sin x xdx = ln tan 2
dx

1

∫ sin 2 x dx = − sin x

dx

π

∫ cos x xdx = ln tan  2 + 4 


dx
∫ sin 2 x xdx = − cot x
dx
∫ cos2 x xdx = tan x

sin( m + n) x sin( m − n) x
+
2( m − n)

∫sin mxsin nxdx = − 2( m+ n)

cos ( m + n) x cos ( m − n) x
−
2( m − n)

∫sin mxcos nxdx = − 2( m + n)

sin ( m + n) x sin ( m − n) x
+
2( m − n)

dx
cos x
1
x
∫ sin 3 x = − 2sin 2 x + 2 ln tan 2

∫ cos mxcos nxdx = 2( m + n)

dx
sin x
1
x π
∫ cos3 x = 2 cos2 x + 2 ln tan  2 + 4 



n
∫ sin x cos xdx = −

1
∫ sin x cos xdx = − 4 cos 2 x
1 3
2
∫ sin x cos xdx = 3 sin x
1
2
3
∫ sin x cos xdx = − 3 cos x
x 1
2
2
∫ sin x cos xdx = 8 − 32 sin 4 x

n
∫ sin x cos xdx =

∫ tan xdx = − ln cos x
sin x
1
dx =
2
cos x
x

∫ cos

sin 2 x
x π 
∫ cos x dx = ln tan  2 + 4  − sin x



∫ tan xdx = tan x − x
∫ cot xdx = ln sin x
2

cos n +1 x
n +1

sin n +1 x
n +1

∫ arcsin xdx = x arcsin x +

1 − x2

∫ arccos xdx = x arccos x −

1 − x2
1

∫ arctan xdx = x arctan x − 2 ln ( x
1

2

∫ arc cot xdx = x arc cot x + 2 ln ( x

2

)

+1

)

+1

m2 ≠ n2
m2 ≠ n2
m2 ≠ n2



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