Tabelas de Integrais Indefinidas

Prévia do material em texto

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
a
a
, 1≠a 
 
 
7. ∫ = ||ln1 xdxx 
 
8. ∫ = |)(|ln)(
)('
xfdx
xf
xf
 
 
 
9. ∫ = a
edxe
ax
ax
 
 
 
 
 
10. ∫ = )ln(a
adxa
x
x
 
 
 
 
11. ∫ −= xxxdxx )ln()ln( 
 
 
12. [ ] 1;)ln()(log)ln()ln(
1)(log ≠−=−=∫ xa
x
xxxxx
a
dxx aa 
 
 
 
 
13. ∫ 





−=





=
+
−−
a
xCotg
aa
x
tg
aax
dx 11
22
11
 
 
 
 
14. ∫ +
−
=





−=
−
−
ax
ax
aa
x
tgh
aax
dx ln
2
11 1
22 
 
 
15. ∫







<>







−−
+−
−
>







=
+
−
0;0;ln
2
1
0;1 1
2
ba
abx
abx
ab
ab
a
abx
tg
ab
bxa
dx
 
 
 
 
16. ∫ 




 +
=
+ b
bxa
bbxa
xdx 2
2 ln2
1
 
 
 
17. 1;)()1(2
32
)(
1
)1(2
1
)( 12122 >+×−
−
+
+
×
−
=
+ ∫∫ −−
m
bxa
dx
am
m
bxamabxa
dx
mmm
 
 
 
18. 1;))((1(2
1
)( 122 >+−
−
=
+∫ −
m
bxambbxa
xdx
mm
 
 
19. 





−=





=
−
−−
∫ a
x
a
x
sen
xa
dx 11
22
cos 
 
 
20. )ln( 22
22
axx
ax
dx ±+=
±
∫ 
 
 
21. ∫ 





=
−
−
x
a
aaxx
dx 1
22
cos
1
 
 
 
22. ∫ 






 ++
−=
± x
xaa
axax
dx 22
22
ln1 
 
 
23. ( )( )∫ ±+±±=± 2222222 ln21 axxaaxxdxax 
 
24. ∫ 






 ++
−+=
+
x
axa
aaxdx
x
ax 2222
22
ln 
 
25. ∫ 





−−=
−
−
x
a
aaxdx
x
ax 122
22
cos 
 
26. ∫ ±=
±
22
22
ax
ax
xdx
 
 
27. ( ) 2/32222
3
1
∫ ±=± axdxaxx 
 
 
28. ( ) ( ) ( )[ ]∫ ±++±±±=± 2242222/3222/322 ln33281 axxaaxxaaxxdxax 
 
 
 
 
29. ( )∫ ±
±
=
± 222
2/322
axa
x
ax
dx
 
30. ( ) ( ) ( ) ( )( )2242222/322222 ln
884
axx
a
ax
xa
ax
xdxaxx ±+−±±=±∫ m 
 
 
31. ∫ 











+−=− −
a
x
senaxaxdxxa 122222
2
1
 
 
 
32. ∫ 







−+
−−=
−
x
xaa
axadx
x
xa 2222
22
ln 
 
 
33. ∫ 







−+
−=
−
x
xaa
a
dx
xax
22
22
ln11 
 
 
34. ∫ −−=
−
22
22
xadx
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
a
dxXxf
2
,
1),( 
 
abyx /)( −= ; 2bacd −= ; cbxaxX ++= 22 
 
 
 
39. ( ) ( )∫∫ −
+
+
++
+
++
+
=+ dxbxax
nm
an
nm
bxaxdxbxax nn
nn
nn 1
1
11
)(
 
 
40. )ln(1 bxa
bbxa
dx
+=
+∫
 
 
41. [ ])ln(12 bxaabxabbxa
xdx
+−+=
+∫
 
 
42. 



+
++=
+∫ bxa
abxa
bbxa
xdx )ln(1)( 22 
 
43. 





+−
+
+−
−
=
+ −−∫ 122 ))(1())(2(
11
)( mmm bxam
a
bxambbxa
xdx
; 3≥m 
 
 
44. ∫ +=+
2)(
3
2 bxa
b
dxbxa 
 
 
45. [ ]∫ ∫ +−++=+ − dxbxaxmabxaxbmdxbxax mmm 12)()32( 2 
 
 
46. ∫∫ +−
−
−
−
+−
=
+ −− )()22(
)32(
)1(
)(
)( 11 bxax
dx
am
bm
xma
bxa
bxax
dx
mmm
; 1≠m 
 
 
47. ∫ ∫ 




 −
=+ zdzz
b
azf
b
dxbxaxf ,2),(
2
; bxaz +=2 
 
 
48. 










 −
+





+−
+
=
+
−
∫ 3
23)(ln
2
1
3
1 1
22
2
222
a
ax
tg
xaxa
xa
axa
dx
 
 
 
49. )cos()( xdxxsen −=∫ 
 
 
50. )()cos( xsendxx =∫ 
 
 
51. ))ln(cos()( xdxxtg −=∫ 
52. ))(ln()(cot xsendxxg =∫ 
 
53. 











+=∫ 22
ln)sec( pixtgdxx 
 
 
54. 











=∫ 2
ln)(cos xtgdxxec 
 
 
55. [ ])()cos(
2
1)(2 xsenxxdxxsen −=∫ 
 
56. ∫∫
−
−
−
+
−
= dxxsen
m
m
m
xsenxdxxsen m
m
m 2
1
)(1)()]cos()( 
 
57. )2(
4
1
2
1)(cos2 xsenxdxx +=∫ 
 
 
58. dxx
m
m
m
xxsendxx m
m
m
∫∫
−
−
−
+= )(cos1)(cos)()(cos 2
1
 
 
59. )()(sec)(cos
2
2 xtgdxxx
dx
∫∫ == 
 
60. ∫∫ −−
−
−
+
−
= )(cos1
2
)(cos)1(
)(
)(cos 21 x
dx
m
m
xm
xsen
x
dx
mmm
; 1>m 
 
 
61. )(cot)(cos)(
2
2 xgdxxecxsen
dx
∫∫ −== 
 
 
62. ∫∫ −−
−
−
+
−
−
= )(1
2
)()1(
)cos(
)( 21 xsen
dx
m
m
xsenm
x
xsen
dx
mmm
; 1>m 
 
 
63. ∫ 





=
± 24)(1
x
tg
xsen
dx
mm
pi
 
 
64. ∫ 





=
+ 2)cos(1
x
tg
x
dx
 
 
 
65. ∫ 





−=
− 2
cot)cos(1
xg
x
dx
 
 
 
66. ∫













>












−
+





−
>












−++





−−+





−
=
+
− 22
22
1
22
22
22
22
22
;22
;
2
2ln1
)(
ba
ba
bxtga
tg
ba
ab
abbxtga
abbxtga
ab
xsenba
dx
 
 
 
 
67. ∫













>












+






−
−
>












−−





−
++





−
−
=
+
− 22
22
1
22
22
22
22
22
;
22
;
2
2ln1
)cos(
ba
ba
x
tgba
tg
ba
ab
baxtgab
baxtgab
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
≠−
−
= ∫∫
−
−
ndxxtg
n
xtgdxxtg n
n
n
 
 
 
72. ∫ = ))(ln()cos()( xtgxxsen
dx
 
 
73. ∫ ∫ >+
−
=
−−
1;)(cos)()(cos)1(
1
)(cos)( 11 mxxsen
dx
xmxxsen
dx
mmm
 
 
 
74. ∫∫
−+−= dxxxmxxdxxsenx mmm )cos()cos()( 1 
 
 
75. ∫∫
−
−= dxxsenxmxsenxdxxx mmm )()()cos( 1 
 
 
76. 211 1)()( xxsenxdxxsen −+=∫ −− 
 
 
77. 211 1)(cos)(cos xxxdxx −−=∫ −− 
 
 
78. ( )211 1ln
2
1)()( xxtgxdxxtg +−=∫ −− 
 
79. ( )211 1ln
2
1)(cot)(cot xxgxdxxg ++=∫ −− 
 
 
80. ( ) ( ) )(122)()( 122121 xsenxxxsenxdxxsen −−− −+−=∫ 
 
 
81. ( ) ( ) )(cos122)(cos)(cos 122121 xxxxxdxx −−− −−−=∫ 
 
 
 
82. dx
x
x
nn
xsenxdxxsenx
nn
n
∫∫
−
+
−
+
=
+−+
−
2
111
1
11
1
1
)()(83. dx
x
x
nn
xxdxxx
nn
n
∫∫
−
+
+
+
=
+−+
−
2
111
1
11
1
1
)(cos)(cos 
 
84.
4
)ln(
2
)ln(
22 x
x
xdxxx −=∫ 
 
85. 1;)1(1)ln( 2
121
≠
+
−
+
=
++
∫ mm
x
m
xdxxx
mMn
m
 
 
 
86. ( ) ( ) ( )∫∫ −−= dxxqxxdxx qqq 1)ln()ln()ln( 
 
 
87. ( ) ( )∫ +=
+
1
)ln()ln( 1
q
xdx
x
x
qq
 
 
 
88. ∫ = ))ln(ln()ln( xdxxx
dx
 
 
 
89. ∫∫ ≠+
−
+
=
−
+
1;))(ln(
11
))(ln())(ln( 1
1
mdxqxx
m
q
m
xxdxxx m
qm
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
ax
ax
 
 
 
93. ∫∫
−
−= dxex
a
exdxex axm
axm
axm 1 ; 0>m 
 
 
94. 1;
1)1( 11 >−+−−= ∫∫ −− mdxx
e
m
a
xm
edx
x
e
m
ax
m
ax
m
ax
 
 
 
95. ∫∫ −= dxx
e
aa
xedxxe
axax
ax 1)ln()ln( 
 
96. ∫ +
−
= 22
))cos()(()(
na
nxnnxsenaedxnxsene
ax
ax
 
 
 
97. ∫ +
+
= 22
))()cos(()cos(
nx
nxsennnxaedxnxe
ax
ax
 
 
 
98. ∫ +−=+
)ln(1 ax
ax
bea
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. ∫ 











=
2
ln)(cos xtghdxxech 
 
 
105.
( )∫ ∫
∫







=
−






=
+






+
=
)(;
1
2
;
11
22
))((
2
22
xsenu
u
du
uf
x
tgz
z
dz
z
zf
dxxsenf 
 
 
106.
( )
∫
∫
∫







=
−
−






=
+






+
−
=
)cos(;
1
2
;
11
12
))(cos(
2
22
2
xu
u
du
uf
x
tgz
z
dz
z
zf
dxxf 
 
 
 
107. ( )∫ ∫
∫







=
−
−






=
+






+
−
+
=
)(;
1
1,
2
;
11
1
,
1
22
))cos(),((
2
2
22
2
2
xsenu
u
du
uuf
x
tgz
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 n
n
n
n
n
n
dxxdxxsen nn
pi
pipi
 
 
 
 
109.









==
+××+×+
−×××
==
+××+×+
−×××
=×
+×××
−×××−×××
=∫
LL
L
L
LL
L
L
K
L
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
nm
nmmm
n
nm
mnnn
m
nm
nm
nm
dxxxsenn
pi
pi
 
 
 
110. ∫
+∞
∞−
−
= pi22/
2
dxe x