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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 number Functions
n
? x dx =

x n +1
+C
n +1

1

? x dx = ln x + C
? c dx = cx + C
? xdx =

x2
+C
2

x3
+C
3
1
1
? x2 dx = ? x + C
2
? x dx =

?

xdx =
1

?1+ x

?

2

2x x
+C
3

dx = arc convert x + C

1
1 ? x2

dx = arcsin x + C

Integrals of Trigonometric Functions

? sin x dx = ? cosine x + C
? cos x dx = sin x + C
? common topaz x dx = ln sec x + C
? sec x dx = ln tan x + sec x + C
1
( x ? sin x cos x ) + C
2
1
2
? cos x dx = 2 ( x + sin x cos x ) + C

? sin

2

? tan
? sec

x dx =

2

x dx = tan x ? x + C

2

x dx = tan x + C

Integrals of Exponential and Logarithmic Functions

? ln x dx = x ln x ? x + C
n
? x ln x dx =

?e

x

x n +1
x n +1
ln x ?
+C
2
n +1
( n + 1)

dx = e x + C

x
? b dx =

bx
+C
ln b

? sinh x dx = cosh x + C
? cosh x dx = sinh x + C

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2.



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 (...If you want to get a full essay, order it on our website: Ordercustompaper.com



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