Table of Integrals

Table of Integrals

BASIC FORMS

(1)

! xndx = 1 xn+1 n+1

(2)

!

1 x

dx

=

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 +

b

dx

=

1 a

ln(ax

+

b)

(6)

!

(x

1 + a)2

dx

=

"1 x+a

! (7)

(x

+

a)n

dx

=

(x

+

a)n

" #$

a 1+n

+

1

x +

n

% &'

,

n ! "1

! (8)

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

(n + 2)(n + 1)

! (9)

dx 1+ x2

=

tan"1 x

! (10)

dx a2 + x2

=

1 tan"1(x / a) a

! (11)

xdx a2 + x2

=

1 ln(a2 2

+ x2 )

! (12)

x 2 dx a2 + x2

=

x " a tan"1(x / a)

! (13)

x 3 dx a2 + x2

=

1 x2 2

"

1 a2 ln(a2 2

+ x2 )

" (14)

(ax2 + bx + c)!1 dx =

2 4ac ! b2

tan!1

# $%

2ax + b & 4ac ! b2 '(

(15)

!

(

x

+

1 a)(x

+

b)

dx

=

b

1 "

a

[ln(a

+

x)

"

ln(b

+

x

)]

,

a!b

(16)

!

(x

x + a)2

dx

=

a

a +

x

+

ln(a

+

x)

! ax2

x + bx

+

c

dx

=

ln(ax2 + bx 2a

+

c)

(17)

!!!!!" a

b 4ac " b2

tan

"1

# $%

2ax + b & 4ac " b2 '(

INTEGRALS WITH ROOTS

" (18)

x ! adx = 2 (x ! a)3/2

3

(19)

!

1 dx = 2 x ? a x?a

(20)

"

1 dx = 2 a ! x a!x

" (21) x x ! adx = 2 a(x ! a)3/2 + 2 (x ! a)5/2

3

5

(22)

!

ax

+

bdx

=

" #$

2b 3a

+

2x 3

% &'

b + ax

! (23)

(ax + b)3/2 dx =

b+

ax

" #$

2b2 5a

+

4bx 5

+

2 ax 2 5

% &'

(24)

!

x dx = 2 (x ? 2a) x ? a

x?a 3

" (25)

a

x !

x

dx

=

!

x

a

!

x

!

a

tan!1

# $%

x a!x& x ! a '(

(26)

!

x dx = x+a

x

x + a " a ln #$ x +

x + a %&

! (27)

x

ax + bdx

=

# $%

"

4b2 15a2

+

2bx 15a

+

2x2 5

& '(

b + ax

!x

ax

+

bdx

=

" #$

bx 4a

+

x 3/2 2

% &'

b + ax

(28)

( ) b2 ln 2 a x + 2 b + ax

!!!!!!!!!!!!!!!!!!!!!!!!!(

4 a 3/2

! x3/2

ax + bdx

=

# $% "

b2 x 8a2

+

bx 3/2 12a

+

x5/2 3

& '(

b + ax

(29)

( ) b3 ln 2 a x + 2 b + ax

"

8a5/2

! ( ) (30)

x2 ? a2 dx = 1 x x2 ? a2 ? 1 a2 ln x + x2 ? a2

2

2

" (31)

a2 ! x2 dx = 1 x 2

a2

!

x2

!

1 2

a2

# tan!1 %

$

x

a2 ! x2 x2 ! a2

& ( '

! (32) x x2 ? a2 = 1 (x2 ? a2 )3/2 3

( ) ! (33)

1 dx = ln x + x2 ? a2

x2 ? a2

?2005 BE Shapiro

Page 1

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" (34)

1 = sin!1 x

a2 ! x2

a

! (35)

x

= x2 ? a2

x2 ? a2

" (36)

x dx = ! a2 ! x2

a2 ! x2

! ( ) (37)

x2 dx = 1 x x2 ? a2 ! 1 ln x + x2 ? a2

x2 ? a2

2

2

" (38)

x2 dx = ! 1 x

a2 ! x2

2

a!

x2

!

1 2

a2

# tan!1 %

$

x

a2 ! x2 x2 ! a2

& ( '

! ax2

+

bx

+

c

!dx

=

" #$

b 4a

+

x 2

% &'

ax2 + bx + c

(39)

!!!!!!!!!!!!!!+

4ac ( b2 8 a 3/2

ln

" #$

2ax + a

b

+

2

ax 2

+

bc

+

c

% &'

! x ax2 + bx + c !dx =

(40)

!!!!!!!!!!!!!!!#$%

x3 3

+

bx 12a

+

8ac " 3b2 24a2

& '(

ax2 + bx + c

!!!!!!!!!!!!!!"

b(4ac " b2 16a5/2

)

ln

# $%

2ax + a

b

+

2

ax 2

+

bc

+

c

& '(

! (41)

1

dx =

ax2 + bx + c

1 a

ln

" #$

2ax + a

b

+

2

ax 2

+

bx

+

c

% &'

!x

dx = 1 ax2 + bx + c

(42)

ax2 + bx + c a

!!!!!"

b 2 a 3/2

ln

# 2ax + $% a

b

+

2

ax 2

+

bx

+

c

& '(

LOGARITHMS

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

(44)

!

ln(ax) x

dx

=

1 2

(ln(ax))2

(45)

!

ln(ax

+

b)dx

=

ax + a

b

ln(ax

+

b)

"

x

! (46)

ln(a2 x2

?

b2

)dx

=

x

ln(a2

x2

?

b2

)

+

2b a

tan"1

# $%

ax & b '(

"

2x

" (47)

ln(a2

!

b2 x2

)dx

=

x ln(a2

!

b2 x2

)

+

2a b

tan!1

# $%

bx a

& '(

!

2x

! (48)

ln(ax2 + bx + c)dx = 1 a

4ac

"

b2

tan"1

# $%

2ax + b & 4ac " b2 '(

( ) !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

"2

x

+

# $%

b 2a

+

x&'(

ln

ax 2

+ bx + c

! (49)

x ln(ax

+

b)dx

=

b 2a

x

"

1 4

x2

+

1 2

# $%

x2

"

b2 a2

& '(

ln(ax

+

b)

" (50)

x

ln(a2

!

b2 x2

)dx

=

!

1 2

x2

+

1 2

# $%

x2

!

a2 b2

& '(

ln(a2

!

bx 2

)

EXPONENTIALS

! (51) eaxdx = 1 eax a

! ( ) (52)

xeaxdx = 1 a

xeax

+

i" 2 a 3/2

erf

i

ax

where

# erf (x) = 2 x e"t2 dt

!0

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

! (54)

xeax

dx

=

# $%

x a

"

1 a2

& '(

eax

! (55) x2exdx = ex (x2 " 2x + 2)

! (56)

x2eax dx

=

eax

# $%

x2 a

"

2x a2

+

2 a3

& '(

! (57) x3exdx = ex (x3 " 3x2 + 6x " 6) ! (58) xneaxdx = ("1)n 1 #[1+ n, "ax] where

a

$ !(a, x) = # t a"1e"t dt x

( ) ! (59) eax2 dx = "i # erf ix a 2a

TRIGONOMETRIC FUNCTIONS

(60) ! sin xdx = " cos x

(61)

!

sin2

xdx

=

x 2

"

1 4

sin

2x

(62)

!

sin3

xdx

=

"

3 4

cos

x

+

1 12

cos

3x

(63) ! cos xdx = sin x

(64)

!

cos2

xdx

=

x 2

+

1 4

sin 2x

(65)

!

cos3

xdx

=

3 4

sin

x

+

1 12

sin 3x

(66)

!

sin

x

cos

xdx

=

"

1 2

cos2

x

?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.

(67)

! sin2

x cos xdx

=

1 4

sin x

"

1 12

sin

3x

(68)

!

sin

x

cos2

xdx

=

"

1 4

cos

x

"

1 12

cos

3x

! (69) sin2 x cos2 xdx = x " 1 sin 4 x 8 32

(70) ! tan xdx = " ln cos x

(71) ! tan2 xdx = "x + tan x ! (72) tan3 xdx = ln[cos x] + 1 sec2 x

2

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

(74) ! sec2 xdx = tan x

(75)

!

sec3

xdx

=

1 2

sec

x

tan

x

+

1 2

ln

|

sec

x

tan

x

|

(76) ! sec x tan xdx = sec x ! (77) sec2 x tan xdx = 1 sec2 x

2

! (78) secn x tan xdx = 1 secn x , n ! 0 n

! xn cos axdx =

(89)

!!!!!!!!!!

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 a

cos

ax

+

1 a2

sin

ax

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

! (93)

x3

sin axdx

=

2

" a2x2 a3

cos ax

+

2 a3

x sin ax

! (94)

xn

sin

xdx

=

"

1 2

(i)n

$%#(n

+

1,

"ix)

"

("1)n

#(n

+

1,

"ix)&'

TRIGONOMETRIC FUNCTIONS WITH eax

(95)

!

ex

sin

xdx

=

1 2

ex

[sin

x

"

cos

x]

! (96)

ebx

sin(ax)dx

=

b2

1 +

a2

ebx

[bsin ax

"

a cos ax]

(97)

!

ex

cos

xdx

=

1 2

ex

[sin

x

+

cos

x]

! (98)

ebx

cos(ax)dx

=

b2

1 +

a2

ebx

[ a sin

ax

+

b cos ax]

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

(80) ! csc2 xdx = " cot x

(81)

!

csc3

xdx

=

"

1 2

cot

x

csc

x

+

1 2

ln

|

csc

x

"

cot

x

|

! (82) cscn x cot xdx = " 1 cscn x , n ! 0 n

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

TRIGONOMETRIC FUNCTIONS WITH xn AND eax

(99)

!

xex

sin

xdx

=

1 2

ex

[cos

x

"

x

cos

x

+

x sin

x]

(100)

!

xex

cos

xdx

=

1 2

ex

[x

cos

x

"

sin

x

+

x sin

x]

HYPERBOLIC FUNCTIONS

(101) ! cosh xdx = sinh x

TRIGONOMETRIC FUNCTIONS WITH xn

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

(85)

!

x cos(ax)dx

=

1 a2

cos ax

+

1 a

x sin ax

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

! (87)

x2

cos axdx

=

2 a2

x cos ax

+

a2x2 " a3

2

sin ax

! xn cos xdx =

(88)

!!!!!!!!!"

1 2

(i

)1+n

$%#(1

+

n,

"ix)

+

("1)n

#(1

+

n,

ix)&'

?2005 BE Shapiro

! (102)

eax

cosh bxdx

=

eax a2 " b2

[a coshbx

"

b sinh bx ]

(103) ! sinh xdx = cosh x

! (104)

eax

sinh bxdx

=

eax a2 " b2

["b cosh bx

+

a sinh bx ]

! (105) ex tanh xdx = ex " 2 tan"1(ex )

(106)

!

tanh axdx

=

1 a

ln

cosh

ax

! cos ax cosh bxdx =

(107)

!!!!!!!!!!

a2

1 +

b2

[ a sin

ax

cosh bx

+

b cos ax sinh bx]

Page 3

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! cos ax sinh bxdx =

(108)

!!!!!!!!!!

a2

1 +

b2

[b

cos

ax

cosh

bx

+

a

sin

ax

sinh

bx]

! sin ax cosh bxdx =

(109)

!!!!!!!!!!

a2

1 +

b2

["a

cos

ax

cosh bx

+

b sin

ax

sinh bx ]

! sin ax sinh bxdx =

(110)

!!!!!!!!!!

a2

1 +

b2

[b

cosh

bx

sin

ax

"

a

cos ax

sinh

bx]

(111)

! sinh ax cosh axdx =

1 4a

["2ax

+

sinh(2ax

)]

! sinh ax cosh bxdx =

(112)

!!!!!!!!!! b2

1 "

a2

[b

cosh bx

sinh

ax

"

a cosh ax sinh bx]

?2005 BE Shapiro

Page 4

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.

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