Tabelas de Integrais Indefinidas

Observação: Em todas as fórmulas, a constante arbitrária é omitida; α,,, cba

representam números reais e qpnm ,,, inteiros positivos. Quando 2a aparece no

integrando, a deve ser tomado como um número positivo, ln( ) pode sempre ser

substituído por ln | | .

1. ∫ = cxcdx

2. ∫ ∫= dxxfcdxxcf )()(

3. ( ) ∫∫ ∫ +=+ dxxgdxxfdxxgxf )()()()(

4. ∫ ∫−= dxxfxgxgxfdxxgxf )()()()()()( ''

5. ∫ ∫−= vduuvudv

6. ∫ +=

+

1

1

a

xdxx

aa , 1≠a

7. ∫ = ||ln1

xdxx

8. ∫ = |)(|ln)(

)('xfdx

xf

xf

9. ∫ =a

edxe

axax

10. ∫ =)ln(a

adxa

xx

11. ∫ −= xxxdxx )ln()ln(

12. [ ] 1;)ln(

)(log)ln()ln(

1)(log ≠−=−=∫ x

a

xxxxxx

adxx aa

13. ∫

−=

=

+

−−

a

xCotg

aa

xtg

aax

dx 11

22

11

14. ∫ +

−=

−=

−

−

ax

ax

aa

xtgh

aax

dxln

2

11 1

22

15. ∫

<>

−−

+−

−

>

=+

−

0;0;ln2

1

0;1 1

2

baabx

abx

ab

aba

abxtg

ab

bxa

dx

16. ∫

+=

+ b

bxa

bbxa

xdx2

2ln

2

1

17. 1;)()1(2

32

)(

1

)1(2

1

)( 12122>

+×

−

−+

+×

−=

+∫∫ −−

mbxa

dx

am

m

bxamabxa

dxmmm

18. 1;))((1(2

1

)( 122>

+−

−=

+∫ −

mbxambbxa

xdxmm

19.

−=

=

−

−−

∫ a

x

a

xsen

xa

dx 11

22cos

20. )ln( 22

22axx

ax

dx±+=

±∫

21. ∫

=

−

−

x

a

aaxx

dx 1

22cos

1

22. ∫

++−=

± x

xaa

axax

dx 22

22ln

1

23. ( )( )∫ ±+±±=±2222222 ln

2

1axxaaxxdxax

24. ∫

++−+=

+

x

axaaaxdx

x

ax 2222

22

ln

25. ∫

−−=

− −

x

aaaxdx

x

ax 12222

cos

26. ∫ ±=±

22

22ax

ax

xdx

27. ( )2/3

2222

3

1∫ ±=± axdxaxx

28. ( ) ( ) ( )[ ]∫ ±++±±±=±2242222/3222/322 ln332

8

1axxaaxxaaxxdxax

29.( )∫

±

±=

±2222/322

axa

x

ax

dx

30. ( ) ( ) ( ) ( )( )224

222

2/322222 ln884

axxa

axxa

axx

dxaxx ±+−±±=±∫ m

31. ∫

+−=−

−

a

xsenaxaxdxxa 122222

2

1

32. ∫

−+−−=

−

x

xaaaxadx

x

xa 2222

22

ln

33. ∫

−+−=

− x

xaa

adx

xax

22

22ln

11

34. ∫ −−=−

22

22xadx

xa

x

35. ∫ ∫ ××=+ duuuautgafadxaxxf )(sec))sec(),((),( 222 ; )(utgax ×=

36. ∫ ∫ ×××=− duutguutgauafadxaxxf )()sec())(),sec((),( 22 ; )sec(uax ×=

37. ∫ ∫ ××−=− duusenusenauafadxxaxf )())(),cos((),( 22 ; )cos(uax ×=

38. ∫ ∫

+−= dy

a

dy

a

byf

adxXxf

2

,1

),(

abyx /)( −= ; 2bacd −= ; cbxaxX ++= 22

39. ( ) ( )∫∫−

+

+++

+++

+=+ dxbxax

nm

an

nm

bxaxdxbxax

nnnn

nn 11

11

)(

40. )ln(1

bxabbxa

dx+=

+∫

41. [ ])ln(1

2bxaabxa

bbxa

xdx+−+=

+∫

42.

+++=

+∫ bxa

abxa

bbxa

xdx)ln(

1

)( 22

43.

+−+

+−

−=

+−−∫ 122 ))(1())(2(

11

)( mmmbxam

a

bxambbxa

xdx; 3≥m

44. ∫ +=+2)(

3

2bxa

bdxbxa

45. [ ]∫ ∫ +−++

=+−

dxbxaxmabxaxbm

dxbxaxmmm 12)(

)32(

2

46. ∫∫+−

−−

−

+−=

+−−

)()22(

)32(

)1(

)(

)( 11bxax

dx

am

bm

xma

bxa

bxax

dxmmm

; 1≠m

47. ∫ ∫

−=+ zdzz

b

azf

bdxbxaxf ,

2),(

2

; bxaz +=2

48.

−+

+−

+=

+

−

∫3

23

)(ln

2

1

3

1 1

22

2

222a

axtg

xaxa

xa

axa

dx

49. )cos()( xdxxsen −=∫

50. )()cos( xsendxx =∫

51. ))ln(cos()( xdxxtg −=∫

52. ))(ln()(cot xsendxxg =∫

53.

+=∫ 22

ln)sec(πx

tgdxx

54.

=∫ 2

ln)(cosx

tgdxxec

55. [ ])()cos(2

1)(2

xsenxxdxxsen −=∫

56. ∫∫−

−−

+−

= dxxsenm

m

m

xsenxdxxsen

mm

m 21

)(1)()]cos(

)(

57. )2(4

1

2

1)(cos2

xsenxdxx +=∫

58. dxxm

m

m

xxsendxx

mm

m

∫∫−

−−

+= )(cos1)(cos)(

)(cos 21

59. )()(sec)(cos

2

2xtgdxx

x

dx∫∫ ==

60. ∫∫ −− −

−+

−=

)(cos1

2

)(cos)1(

)(

)(cos 21 x

dx

m

m

xm

xsen

x

dxmmm

; 1>m

61. )(cot)(cos)(

2

2xgdxxec

xsen

dx∫∫ −==

62. ∫∫ −−−

−+

−

−=

)(1

2

)()1(

)cos(

)( 21 xsen

dx

m

m

xsenm

x

xsen

dxmmm

; 1>m

63. ∫

=

± 24)(1

xtg

xsen

dxmm

π

64. ∫

=

+ 2)cos(1

xtg

x

dx

65. ∫

−=

− 2cot

)cos(1

xg

x

dx

66. ∫

>

−

+

−

>

−++

−−+

−

=+

− 22

22

1

22

22

22

22

22

;22

;

2

2ln

1

)(

baba

bx

tga

tgba

ab

abbx

tga

abbx

tga

ab

xsenba

dx

67. ∫

>

+

−

−

>

−−

−

++

−

−

=+

− 22

22

1

22

22

22

22

22

;22

;

2

2ln

1

)cos(

baba

xtgba

tgba

ab

bax

tgab

bax

tgab

ab

xba

dx

68. 22;)(2

)(

)(2

)()()( nm

nm

xnmsen

nm

xnmsendxmxsennxsen ≠

+

+−

−

−=×∫

69. 22;)(2

)cos(

)(2

)cos()cos()( nm

nm

xnm

nm

xnmdxmxnxsen ≠

+

+−

−

−=×∫

70. 22;)(2

)(

)(2

)()cos()cos( nm

nm

xnmsen

nm

xnmsendxmxnx ≠

+

++

−

−=×∫

71. 1;)(1

)()( 2

1

≠−−

= ∫∫−

−

ndxxtgn

xtgdxxtg

nn

n

72. ∫ = ))(ln()cos()(

xtgxxsen

dx

73. ∫ ∫ >+−

=−−

1;)(cos)()(cos)1(

1

)(cos)( 11m

xxsen

dx

xmxxsen

dxmmm

74. ∫∫−

+−= dxxxmxxdxxsenx mmm )cos()cos()( 1

75. ∫∫−

−= dxxsenxmxsenxdxxx mmm )()()cos( 1

76. 211 1)()( xxsenxdxxsen −+=∫−−

77. 211 1)(cos)(cos xxxdxx −−=∫−−

78. ( )211 1ln2

1)()( xxtgxdxxtg +−=∫

−−

79. ( )211 1ln2

1)(cot)(cot xxgxdxxg ++=∫

−−

80. ( ) ( ) )(122)()( 122121xsenxxxsenxdxxsen

−−−−+−=∫

81. ( ) ( ) )(cos122)(cos)(cos 122121xxxxxdxx

−−−−−−=∫

82. dxx

x

nn

xsenxdxxsenx

nnn

∫∫−+

−+

=

+−+

−

2

1111

11

1

1

)()(

83. dxx

x

nn

xxdxxx

nnn

∫∫−+

++

=

+−+

−

2

1111

11

1

1

)(cos)(cos

84.4

)ln(2

)ln(22 x

xx

dxxx −=∫

85. 1;)1(1

)ln(2

121

≠+

−+

=

++

∫ mm

x

m

xdxxx

mMnm

86. ( ) ( ) ( )∫∫−

−= dxxqxxdxxqqq 1

)ln()ln()ln(

87.( ) ( )∫

+=

+

1

)ln()ln(1

q

xdx

x

xqq

88. ∫ = ))ln(ln()ln(

xdxxx

dx

89. ∫∫ ≠+

−+

=−

+

1;))(ln(11

))(ln())(ln( 1

1

mdxqxxm

q

m

xxdxxx

mqm

qm

90. ∫ −= ))cos(ln(2

1))(ln(

2

1))(ln( xxxsenxdxxsen

91. ∫ += ))cos(ln(2

1))(ln(

2

1))cos(ln( xxxsenxdxx

92. )1(2

−=∫ axa

edxex

axax

93. ∫∫−

−= dxexa

exdxex

axmaxm

axm 1 ; 0>m

94. 1;1)1( 11

>−

+−

−= ∫∫ −−mdx

x

e

m

a

xm

edx

x

em

ax

m

ax

m

ax

95. ∫∫ −= dxx

e

aa

xedxxe

axaxax 1)ln(

)ln(

96. ∫+

−=

22

))cos()(()(

na

nxnnxsenaedxnxsene

axax

97. ∫+

+=

22

))()cos(()cos(

nx

nxsennnxaedxnxe

axax

98. ∫ +−=+

)ln(1 ax

axbea

aqa

x

bea

dx

99. ∫ = )cosh()( xdxxsenh

100. ∫ = )()cosh( xsenhdxx

101. ∫ = )cosh(ln)( xdxxtgh

102. ∫ = )(ln)(cot xsenhdxxgh

103. ∫−−

== ))((2)(sec 11 xsenhtgetgdxxh x

104. ∫

=

2ln)(cos

xtghdxxech

105.

( )∫

∫

∫

=−

=

+

+=

)(;1

2;

11

22

))((

2

22

xsenuu

duuf

xtgz

z

dz

z

zf

dxxsenf

106.

( )∫

∫

∫

=−

−

=

+

+

−

=

)cos(;1

2;

11

12

))(cos(

2

22

2

xuu

duuf

xtgz

z

dz

z

zf

dxxf

107.

( )∫

∫

∫

=−

−

=

+

+

−

+=

)(;1

1,

2;

11

1,

1

22

))cos(),((

2

2

22

2

2

xsenuu

duuuf

xtgz

z

dz

z

z

z

zf

dxxxsenf

108.

=−

×××

=×−

×××

== ∫∫KL

KL

;7;5;3;1

5

4

3

2

;6;4;2;2

1

4

3

2

1

)(cos)(

2/

0

2/

0 nn

n

nn

n

dxxdxxsennn

πππ

109.

==+××+×+

−×××

==+××+×+

−×××

=×+×××

−×××−×××

=∫

LLL

L

LLL

L

KL

LL

;7;5;3,;3;2;1;)()3()1(

)1(42

;3;2;1,;7;5;3;)()3()1(

)1(42

;4;2,;2)(42

)1(31)1(31

)(cos)(

2/

0

2

nmnmmm

n

nmmnnn

m

nmnm

nm

dxxxsenn

π

π

110. ∫+∞

∞−

−= π22/

2

dxe x