Table of Integrals - UMD

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

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 = 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

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.

! 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.

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download