Baixe tabela de integrais e derivadas e outras Resumos em PDF para Cálculo Diferencial e Integral, somente na Docsity!
©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
Table of Integrals
BASIC FORMS
(1)! x n^ dx = 1
n + 1 x^
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)^1
( x + a ) 2
! dx =^ "^1
x + a
(7)! ( x + a ) n^ dx = ( x + a ) n^ a
1+ n
#$^
(^) &', n! " 1
(8)! x ( x + a ) n^ dx = ( x^ +^ a )
1 + n (^) ( nx + x " a ) ( n + 2)( n + 1)
(9) dx
! 1 + x 2
= tan"^1 x
(10) dx
! a 2 + x 2
a
tan"^1 ( x / a )
(11) xdx
! a 2 + x 2
ln( a^2 + x^2 )
(12) x
(^2) dx
! a^2 + x^2 =^ x^ "^ a^ tan
" (^1) ( x / a )
(13) x
(^3) dx
! a^2 + x^2 =^
2 x
2 a
(^2) ln( a (^2) + x (^2) )
(14) " ( ax^2 + bx + c )!^1 dx = 2
4 ac! b^2
tan!^1 2 ax^ +^ b 4 ac! b^2
(15) ! ( x + a )(^1 x + b ) dx = b^1 " a [ ln( a + x ) " ln( b + x )], a! b
(16) x
! ( x + a ) 2 dx^ =
a a + x +^ ln( a^ +^ x )
x ax^2 + bx + c
! dx =^ ln( ax
(^2) + bx + c ) 2 a !!!!!" b a 4 ac " b^2
tan"^1 2 ax^ +^ b 4 ac " b^2
INTEGRALS WITH ROOTS
(18) " x! adx = 2
3 ( x^!^ a )^
3/
(19)^1
x ± a
! dx =^2 x^ ±^ a
(20)^1
a! x
" dx =^2 a^!^ x
(21) " x x! adx = 2
3 a ( x^!^ a )^
5 ( x^!^ a )^
5/
(22)! ax + bdx = 2 b
3 a
#$^
&'^ b^ +^ ax
(23)! ( ax + b ) 3/2^ dx = b + ax^2 b
2 5 a +^
4 bx 5 +^
2 ax^2 5
(24) x
! x ± a
dx = 2 3
( x ± 2 a ) x ± a
(25) x a! x
" dx^ =^!^ x^ a^!^ x!^ a^ tan!^1 x^ a^!^ x
x! a
(26) x x + a
! dx^ =^ x^ x^ +^ a "^ a^ ln^ #$ x^ +^ x^ +^ a %&
(27)! x ax + bdx = " 4 b
2 15 a^2 +^
2 bx 15 a +^
2 x^2 5
'(^ b^ +^ ax
! x^ ax^ +^ bdx =^ b^ x
4 a
3/ 2
&'^
b + ax
b^2 ln 2( a x + 2 b + ax )
4 a 3/
! x 3/2^ ax^ +^ bdx =^ "^ b
(^2) x 8 a^2
3/ 12 a
5/ 3
'(^
b + ax
b^3 ln 2( a x + 2 b + ax )
8 a 5/
(30)! x^2 ± a^2 dx = 1
x x^2 ± a^2 ± 1 2
a^2 ln ( x + x^2 ± a^2 )
(31) " a^2! x^2 dx = 12 x a^2! x^2! 12 a^2 tan!^1 x^ a
(^2)! x 2 x^2! a^2
(32)! x x^2 ± a^2 = 1
( x^2 ± a^2 ) 3/
(33)^1
x^2 ± a^2
! dx^ =^ ln^ (^ x^ +^ x^2 ±^ 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
(34)^1
" a 2! x 2
= sin!^1 x a
(35) x x^2 ± a^2
! =^ x^2 ±^ a^2
(36) x
" a 2! x 2
dx =! a^2! x^2
(37) x
2
! x 2 ± a 2
dx = 1 2
x x^2 ± a^2! 1 2
ln ( x + x^2 ± a^2 )
(38) x
2
" a 2! x 2
dx =! 1 2
x a! x^2! 1 2
a^2 tan!^1 x^ a
(^2)! x 2 x^2! a^2
! ax^2 +^ bx^ +^ c! dx^ =^ b
4 a
#$^
&'^ ax
(^2) + bx + c
!!!!!!!!!!!!!!+ 4 ac^ (^ b
2 8 a 3/^
ln 2 ax^ +^ b a
"#$ + 2 ax (^2) + bc + c %&'
! x^ ax^2 +^ bx^ +^ c^! dx =
!!!!!!!!!!!!!!! x
3 3
2 24 a^2
'(^
ax^2 + bx + c
!!!!!!!!!!!!!!" b (4 ac^ "^ b
16 a 5/^
ln 2 ax^ +^ b a
#$% + 2 ax (^2) + bc + c &'(
(41)^1
! ax 2 + bx + c
dx = 1 a
ln 2 ax^ +^ b a
" + 2 ax (^2) + bx + c #$^
x
! ax 2 + bx + c
dx = 1 a
ax^2 + bx + c
!!!!!" (^2) ab 3/2 ln 2 ax^ +^ b a
+ 2 ax (^2) + bx + c
$%^
LOGARITHMS
(43)! ln xdx = x ln x " x
(44) ln( ax ) x
! dx =^1
(ln( ax ))^2
(45)! ln( ax + b ) dx = ax^ +^ b
a
ln( ax + b ) " x
(46)! ln( a^2 x^2 ± b^2 ) dx = x ln( a^2 x^2 ± b^2 ) + 2 b
a
tan"^1 ax b
$%^
'( "^2 x
(47) " ln( a^2! b^2 x^2 ) dx = x ln( a^2! b^2 x^2 ) + 2 a
b
tan!^1 bx a
$%^
'(!^2 x
! ln( ax^2 +^ bx^ +^ c ) dx =^1
a
4 ac " b^2 tan"^1 2 ax^ +^ b 4 ac " b^2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!" 2 x + b 2 a
#$% + x &'( ln ( ax 2 + bx + c )
(49)! x ln( ax + b ) dx = b
2 a
x " 1 4
x^2 + 1 2
x^2 " b
2 a^2
'(^
ln( ax + b )
(50) " x ln( a^2! b^2 x^2 ) dx =! 1
x^2 + 1 2
x^2! a
2 b^2
'(^
ln( a^2! bx^2 )
EXPONENTIALS
(51)! e ax^ dx = 1
a
e ax
(52)! xe ax^ dx = 1
a
xe ax^ + i^ " 2 a 3/^
erf ( i ax ) where
erf ( x ) = 2 ! 0 e " t^2^ dt
x
(53)! xe x^ dx = ( x " 1) e x
(54)! xe ax^ dx = x
a
a^2
$%^
'(^ e^
ax
(55)! x^2 e x^ dx = e x^ ( x^2 " 2 x + 2)
(56)! x^2 e ax^ dx = e ax^ x
2 a
" 2 x a^2
a^3
(57)! x^3 e x^ dx = e x^ ( x^3 " 3 x^2 + 6 x " 6)
(58)! x n^ e ax^ dx = ( " 1 ) n^^1
a
#[1+ n , " ax ]where
!( a , x ) = (^) xt a "^1 e " t^ dt
(59) e ax
2
!^ dx =^ " i^
2 a
erf ( ix a )
TRIGONOMETRIC FUNCTIONS
(60)! sin xdx = " cos x
(61)! sin 2 xdx = x
2 "^
4 sin 2 x
(62)! sin 3 xdx = " 3
4 cos^ x^ +^
12 cos 3 x
(63)! cos xdx = sin x
(64)! cos 2 xdx = x
2 +^
4 sin 2 x
(65)! cos 3 xdx = 4 3 sin x + 12 1 sin 3 x
(66)! sin x cos xdx = " 1
cos 2 x
©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
! cos^ ax^ sinh^ bxdx =
a^2 + b^2
[ b cos ax cosh bx + a sin ax sinh bx ]
! sin^ ax^ cosh^ bxdx =
a^2 + b^2
[" a cos ax cosh bx + b sin ax sinh bx ]
! sin^ ax^ sinh^ bxdx =
a^2 + b^2
[ b cosh bx sin ax " a cos ax sinh bx ]
(111)! sinh ax cosh axdx = 1
4 a
[ "^2 ax^ +^ sinh(2 ax )]
! sinh^ ax^ cosh^ bxdx =
b^2 " a^2
[ b^ cosh^ bx^ sinh^ ax^ "^ a^ cosh^ ax^ sinh^ bx ]
