©2005 BE Shapiro Page 1 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 BASIC FORMS

(1) xndx! =

1

n +1xn+1

(2) 1

xdx! = ln x

(3) udv! = uv " vdu!

(4) u(x) !v (x)dx" = u(x)v(x)# v(x) !u (x)dx"

RATIONAL FUNCTIONS

(5) 1

ax + bdx! =

1

aln(ax + b)

(6) 1

(x + a)2dx! =

"1

x + a

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

n a

1+n+

x

1+ n

"#$

%&'

, n ! "1

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

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

(n + 2)(n +1)

(9) dx

1+ x2! = tan

"1x

(10) dx

a2+ x

2! =1

atan

"1(x / a)

(11) xdx

a2+ x

2! =1

2ln(a

2+ x

2)

(12) x2dx

a2+ x

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

(13) x3dx

a2+ x

2! =1

2x2 "1

2a2ln(a

2+ x

2)

(14) (ax2+ bx + c)

!1dx" =

2

4ac ! b2tan

!1 2ax + b

4ac ! b2#

$%&

'(

(15) 1

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

1

b " aln(a + x)" ln(b + x)[ ] , a ! b

(16) x

(x + a)2dx =!

a

a + x+ ln(a + x)

(17)

x

ax2+ bx + c

dx! =ln(ax

2+ bx + c)

2a

!!!!!"b

a 4ac " b2tan

"1 2ax + b

4ac " b2#

$%&

'(

INTEGRALS WITH ROOTS

(18) x ! adx" =2

3(x ! a)3/2

(19) 1

x ± adx! = 2 x ± a

(20) 1

a ! xdx" = 2 a ! x

(21) x x ! adx" =2

3a(x ! a)3/2 +

2

5(x ! a)5/2

(22) ax + bdx! =2b

3a+2x

3

"#$

%&'

b + ax

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

2b2

5a+4bx

5+2ax

2

5

"#$

%&'

(24) x

x ± a! dx =

2

3x ± 2a( ) x ± a

(25) x

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

x ! a

#

$%&

'(

(26) x

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

$%&

(27) x ax + bdx! = "4b

2

15a2+2bx

15a+2x

2

5

#$%

&'(

b + ax

(28) x ax + bdx! =

b x

4a+x3/2

2

"

#$%

&'b + ax

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

4a3/2

(29) x3/2

ax + bdx! = "b2x

8a2

+bx

3/2

12a+x5/2

3

#

$%&

'(b + ax

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

8a5/2

(30) x2± a

2

! dx =1

2x x

2± a

2±1

2a2ln x + x

2± a

2( )

(31) a2 ! x2" dx =

1

2x a

2 ! x2 !1

2a2tan

!1 x a2 ! x2

x2 ! a2

#

$%

&

'(

(32) x x2± a

2

! =1

3(x

2± a

2)3/2

(33) 1

x2± a

2

dx = ln x + x2± a

2( )!

©2005 BE Shapiro Page 2 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.

(34) 1

a2 ! x2

" = sin!1 x

a

(35) x

x2± a

2

= x2± a

2

!

(36) x

a2 ! x2

" dx = ! a2 ! x2

(37)

x2

x2± a

2! dx =1

2x x

2± a

2!1

2ln x + x

2± a

2( )

(38) x2

a2 ! x2

" dx = !1

2x a ! x2 !

1

2a2tan

!1 x a2 ! x2

x2 ! a2

#

$%

&

'(

(39) ax

2+ bx + c! !dx =

b

4a+x

2

"#$

%&'

ax2+ bx + c

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

8a3/2

ln2ax + b

a+ 2 ax

2+ bc + c

"#$

%&'

(40)

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

#$%

&'(

(41) 1

ax2+ bx + c

! dx =1

aln2ax + b

a+ 2 ax

2+ bx + c

"

#$

%

&'

(42)

x

ax2+ bx + c

! dx =1

aax

2+ bx + c

!!!!!"b

2a3/2ln2ax + b

a+ 2 ax

2+ bx + c

#

$%

&

'(

LOGARITHMS

(43) ln x! dx = x ln x " x

(44) ln(ax)

xdx! =

1

2ln(ax)( )

2

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

aln(ax + b)" x

(46) ln(a2x2± b

2! )dx = x ln(a2x2± b

2)+2b

atan

"1 ax

b

#$%

&'(" 2x

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

2 ! b2x2 )+2a

btan

!1 bx

a

#$%

&'(! 2x

(48) ln(ax

2+ bx + c)dx! =

1

a4ac " b2 tan"1 2ax + b

4ac " b2#

$%&

'(

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

2a+ x

#$%

&'(ln ax

2+ bx + c( )

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

2ax "

1

4x2+1

2x2 "

b2

a2

#$%

&'(ln(ax + b)

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

1

2x2+1

2x2 !

a2

b2

#$%

&'(ln(a

2 ! bx2 )

EXPONENTIALS

(51) eaxdx! =

1

aeax

(52) xeaxdx! =

1

axe

ax+i "

2a3/2erf i ax( ) where

erf (x) = 2

!e" t2dt

0

x

#

(53) xexdx! = (x "1)ex

(54) xeaxdx! =

x

a"1

a2

#$%

&'(eax

(55) x2exdx! = e

x(x

2 " 2x + 2)

(56) x2eaxdx! = e

ax x2

a"2x

a2+2

a3

#$%

&'(

(57) x3exdx! = e

x(x

3 " 3x2 + 6x " 6)

(58) xneaxdx! = "1( )

n 1

a#[1+ n,"ax] where

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

x

#

$

(59) eax2

dx! = "i#

2 aerf ix a( )

TRIGONOMETRIC FUNCTIONS

(60) sin xdx! = " cos x

(61) sin2xdx! =

x

2"1

4sin2x

(62) sin3xdx! = "

3

4cos x +

1

12cos3x

(63) cos xdx! = sin x

(64) cos2xdx! =

x

2+1

4sin2x

(65) cos3xdx! =

3

4sin x +

1

12sin 3x

(66) sin x cos xdx! = "1

2cos

2x

©2005 BE Shapiro Page 3 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.

(67) sin2x cos xdx! =

1

4sin x "

1

12sin 3x

(68) sin x cos2xdx! = "

1

4cos x "

1

12cos3x

(69) sin2x cos

2xdx! =

x

8"1

32sin 4x

(70) tan xdx! = " ln cos x

(71) tan2xdx! = "x + tan x

(72) tan3xdx! = ln[cos x]+

1

2sec

2x

(73) sec xdx! = ln | sec x + tan x |

(74) sec2xdx! = tan x

(75) sec3xdx! =

1

2sec x tan x +

1

2ln | sec x tan x |

(76) sec x tan xdx! = sec x

(77) sec2x tan xdx! =

1

2sec

2x

(78) secnx tan xdx! =

1

nsec

nx , n ! 0

(79) csc xdx! = ln | csc x " cot x |

(80) csc2xdx =! " cot x

(81) csc3xdx =! "

1

2cot x csc x +

1

2ln | csc x " cot x |

(82) cscnx cot xdx! = "

1

ncsc

nx , n ! 0

(83) sec x csc xdx! = ln tan x

TRIGONOMETRIC FUNCTIONS WITH xn

(84) x cos xdx! = cos x + x sin x

(85) x cos(ax)dx! =1

a2cosax +

1

ax sinax

(86) x2cos xdx! = 2x cos x + (x

2 " 2)sin x

(87) x2cosaxdx! =

2

a2x cosax +

a2x2 " 2

a3

sinax

(88) xncos xdx! =

!!!!!!!!!"1

2i( )1+n

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

#(1+ n,ix)$%

&'

(89) xncosaxdx! =

!!!!!!!!!!1

2(ia)

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

(90) x sin xdx! = "x cos x + sin x

(91) x sin(ax)dx! = "x

acosax +

1

a2sinax

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

(93) x3sinaxdx! =

2 " a2x2

a3

cosax +2

a3x sinax

(94) xnsin xdx! = "

1

2(i)

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

TRIGONOMETRIC FUNCTIONS WITH eax

(95) exsin xdx! =

1

2exsin x " cos x[ ]

(96) ebxsin(ax)dx! =

1

b2+ a

2ebxbsinax " acosax[ ]

(97) excos xdx! =

1

2exsin x + cos x[ ]

(98) ebxcos(ax)dx! =

1

b2+ a

2ebxasinax + bcosax[ ]

TRIGONOMETRIC FUNCTIONS WITH xn AND eax

(99) xexsin xdx! =

1

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

(100) xexcos xdx! =

1

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

HYPERBOLIC FUNCTIONS

(101) cosh xdx! = sinh x

(102) eaxcoshbxdx! =

eax

a2 " b2

acoshbx " bsinhbx[ ]

(103) sinh xdx! = cosh x

(104) eaxsinhbxdx! =

eax

a2 " b2

"bcoshbx + asinhbx[ ]

(105) extanh xdx! = e

x " 2 tan"1(e

x)

(106) tanhaxdx! =1

aln coshax

(107) cosax coshbxdx! =

!!!!!!!!!!1

a2+ b

2asinax coshbx + bcosax sinhbx[ ]

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(108) cosax sinhbxdx! =

!!!!!!!!!!1

a2+ b

2bcosax coshbx + asinax sinhbx[ ]

(109) sinax coshbxdx! =

!!!!!!!!!!1

a2+ b

2"acosax coshbx + bsinax sinhbx[ ]

(110) sinax sinhbxdx! =

!!!!!!!!!!1

a2+ b

2bcoshbx sinax " acosax sinhbx[ ]

(111) sinhax coshaxdx! =1

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

(112) sinhax coshbxdx! =

!!!!!!!!!!1

b2 " a2

bcoshbx sinhax " acoshax sinhbx[ ]