©2005 BE Shapiro. This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose.

Table of Integrals integral-table.com

Basic Forms

xndx! =

1

n +1xn+1

1

xdx! = ln x

udv! = uv " vdu!

Integrals of Rational Functions

1

ax + bdx! =

1

aln(ax + b)

1

(x + a)2dx! =

"1

x + a

(x + a)ndx! = (x + a)

n a

1+n+

x

1+ n

"#$

%&'

, n ! "1

x(x + a)ndx! =

(x + a)1+n(nx + x " a)

(n + 2)(n +1)

dx

1+ x2! = tan

"1x

dx

a2+ x

2! =1

atan

"1(x / a)

xdx

a2+ x

2! =1

2ln(a

2+ x

2)

x2dx

a2+ x

2! = x " a tan"1(x / a)

x3dx

a2+ x

2! =1

2x2 "

1

2a2ln(a

2+ x

2)

(ax2+ bx + c)

!1dx" =

2

4ac ! b2tan

!1 2ax + b

4ac ! b2#

$%&

'(

1

(x + a)(x + b)dx =!

l n(a + x) " ln(b + x)

b " a,!a # b ,

x

(x + a)2dx =!

a

a + x+ ln(a + x)

x

ax2+ bx + c

dx! =ln(ax

2+ bx + c)

2a

!!!!!"b

a 4ac " b2tan

"1 2ax + b

4ac " b2#

$%&

'(

Integrals Involving Roots

x ! adx" =2

3(x ! a)3/2

1

x ± adx! = 2 x ± a

1

a ! xdx" = 2 a ! x

x x ! adx" =2

3a(x ! a)3/2 +

2

5(x ! a)5/2

ax + bdx! =2b

3a+2x

3

"#$

%&'

b + ax

(ax + b)3/2dx! = b + ax

2b2

5a+4bx

5+2ax

2

5

"#$

%&'

x

x ± a! dx =

2

3x ± 2a( ) x ± a

x

a ! xdx = ! x a ! x" ! a tan!1 x a ! x

x ! a

#

$%&

'(

x

x + adx = x x + a! " a ln x + x + a#

$%&

x ax + bdx! = "4b

2

15a2+2bx

15a+2x

2

5

#$%

&'(

b + ax

x ax + bdx! =b x

4a+x3/2

2

"

#$%

&'b + ax

!!!!!!!!!!!!!!!!!!!!!!!!!(b2ln 2 a x + 2 b + ax( )

4a3/2

x3/2

ax + bdx! = "b2x

8a2

+bx

3/2

12a+x5/2

3

#

$%&

'(b + ax

"b3ln 2 a x + 2 b + ax( )

8a5/2

x2± a

2

! dx =1

2x x

2± a

2±1

2a2ln x + x

2± a

2( )

a2 ! x2" dx =

1

2x a

2 ! x2 !1

2a2tan

!1 x a2 ! x2

x2 ! a2

#

$%

&

'(

x x2± a

2

! =1

3(x

2± a

2)3/2

1

x2± a

2

dx = ln x + x2± a

2( )!

1

a2 ! x2

" = sin!1 x

a

x

x2± a

2

= x2± a

2

!

x

a2 ! x2

" dx = ! a2 ! x2

x2

x2± a

2! dx =1

2x x

2± a

2!1

2ln x + x

2± a

2( )

x2

a2 ! x2

" dx = !1

2x a ! x2 !

1

2a2tan

!1 x a2 ! x2

x2 ! a2

#

$%

&

'(

ax2+ bx + c! !dx =

b

4a+x

2

"#$

%&'

ax2+ bx + c

!!!!!!!!!!!!!!+4ac ( b2

8a3/2

ln2ax + b

a+ 2 ax

2+ bc + c

"#$

%&'

x ax2+ bx + c !dx! =

!!!!!!!!!!!!!!!x3

3+bx

12a+8ac " 3b2

24a2

#$%

&'(

ax2+ bx + c

!!!!!!!!!!!!!!"b(4ac " b2 )16a

5/2ln

2ax + b

a+ 2 ax

2+ bc + c

#$%

&'(

1

ax2+ bx + c

! dx =1

aln2ax + b

a+ 2 ax

2+ bx + c

"

#$

%

&'

x

ax2+ bx + c

! dx =1

aax

2+ bx + c

!!!!!"b

2a3/2ln2ax + b

a+ 2 ax

2+ bx + c

#

$%

&

'(

Integrals Involving Logarithms

ln x! dx = x ln x " x

ln(ax)

xdx! =

1

2ln(ax)( )

2

ln(ax + b)! dx =ax + b

aln(ax + b) " x

ln(a2x2± b

2! )dx = x ln(a2x2± b

2) +2b

atan

"1 ax

b

#$%

&'(" 2x

ln(a2 ! b2x2" )dx = x ln(a

2 ! b2x2 ) +2a

btan

!1 bx

a

#$%

&'(! 2x

ln(ax2+ bx + c)dx! =

1

a4ac " b2 tan"1 2ax + b

4ac " b2#

$%&

'(

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!"2x +b

2a+ x

#$%

&'(ln ax

2+ bx + c( )

x ln(ax + b)dx! =b

2ax "

1

4x2+1

2x2 "

b2

a2

#$%

&'(ln(ax + b)

x ln(a2 ! b2x2 )dx" = !

1

2x2+1

2x2 !

a2

b2

#$%

&'(ln(a

2 ! bx2 )

©2005 BE Shapiro. This document may not be reproduced, posted or published without permission. The copyright holder makes no representation about the accuracy, correctness, or suitability of this material for any purpose.

Integrals Involving Exponentials

eaxdx! =

1

aeax

xeaxdx! =

1

axe

ax+i "

2a3/2erf i ax( ) , where

erf (x) =2

!e" t2dt

0

x

#

xexdx! = (x "1)ex

xeaxdx! =

x

a"1

a2

#$%

&'(eax

x2exdx! = e

x(x

2 " 2x + 2)

x2eaxdx! = e

ax x2

a"2x

a2+2

a3

#$%

&'(

x3exdx! = e

x(x

3 " 3x2 + 6x " 6)

xneaxdx! = "1( )

n 1

a#[1+ n,"ax] where

!(a, x) = ta"1e" tdt

x

#

$

eax2

dx! = "i#

2 aerf ix a( )

Integrals Involving Trigonometric Functions

sin xdx! = " cos x

sin2xdx! =

x

2"1

4sin 2x

sin3xdx! = "

3

4cos x +

1

12cos 3x

cos xdx! = sin x

cos2xdx! =

x

2+1

4sin 2x

cos3xdx! =

3

4sin x +

1

12sin 3x

sin x cos xdx! = "1

2cos

2x

sin2x cos xdx! =

1

4sin x "

1

12sin 3x

sin x cos2xdx! = "

1

4cos x "

1

12cos 3x

sin2x cos

2xdx! =

x

8"1

32sin 4x

tan xdx! = " ln cos x

tan2xdx! = "x + tan x

tan3xdx! = ln[cos x]+

1

2sec

2x

sec xdx! = ln | sec x + tan x |

sec2xdx! = tan x

sec3xdx! =

1

2sec x tan x +

1

2ln | sec x tan x |

sec x tan xdx! = sec x

sec2x tan xdx! =

1

2sec

2x

secnx tan xdx! =

1

nsec

nx , n ! 0

csc xdx! = ln | csc x " cot x |

csc2xdx =! " cot x

csc3xdx =! "

1

2cot x csc x +

1

2ln | csc x " cot x |

cscnx cot xdx! = "

1

ncsc

nx , n ! 0

sec x csc xdx! = ln tan x

Integrals Involving Trigonometric Functions and xn

x cos xdx! = cos x + x sin x

x cos(ax)dx! =1

a2cosax +

1

ax sinax

x2cos xdx! = 2x cos x + (x

2 " 2)sin x

x2cosaxdx! =

2

a2x cosax +

a2x2 " 2

a3

sinax

xncos xdx! =

!!!!!!!!!"1

2i( )1+n

#(1+ n,"ix) + "1( )n

#(1+ n, ix)$%

&'

xncosaxdx! =

!!!!!!!!!!1

2(ia)

1"n("1)n#(1+ n,"iax) " #(1+ n, iax)$% &'

x sin xdx! = "x cos x + sin x

x sin(ax)dx! = "x

acosax +

1

a2sinax

x2sin xdx! = (2 " x2 )cos x + 2x sin x

x2sinaxdx! =

2 " a2x2

a3

cosax +2

a3x sinax

xnsin xdx! = "

(i)n

2#(n +1,"ix) " ("1)n#(n +1,"ix)$% &'

Integrals Involving both Trigonometric and Exponential Functions

exsin xdx! =

1

2exsin x " cos x[ ]

ebxsin(ax)dx! =

1

b2+ a

2ebxb sinax " a cosax[ ]

excos xdx! =

1

2exsin x + cos x[ ]

ebxcos(ax)dx! =

1

b2+ a

2ebxa sinax + b cosax[ ]

xexsin xdx! =

1

2excos x " x cos x + x sin x[ ]

xexcos xdx! =

1

2exx cos x " sin x + x sin x[ ]

Integrals Involving Hyperbolic Functions

cosh xdx! = sinh x

eaxcoshbxdx! =

eax

a2 " b2

a coshbx " b sinhbx[ ]

sinh xdx! = cosh x

eaxsinhbxdx! =

eax

a2 " b2

"b coshbx + a sinhbx[ ]

extanh xdx! = e

x " 2 tan"1(e

x)

tanhaxdx! =1

aln coshax

cosax coshbxdx! =

!!!!!!!!!!1

a2+ b

2a sinax coshbx + b cosax sinhbx[ ]

cosax sinhbxdx! =

!!!!!!!!!!1

a2+ b

2b cosax coshbx + a sinax sinhbx[ ]

sinax coshbxdx! =

!!!!!!!!!!1

a2+ b

2"a cosax coshbx + b sinax sinhbx[ ]

sinax sinhbxdx! =

!!!!!!!!!!1

a2+ b

2b coshbx sinax " a cosax sinhbx[ ]

sinhax coshaxdx! =1

4a"2ax + sinh(2ax)[ ]

sinhax coshbxdx! =

!!!!!!!!!!1

b2 " a2

b coshbx sinhax " a coshax sinhbx[ ]