18.B: Tabela de integrais
Integrais básicos
1. ∫undu=un+1n+1+C,n≠−1
2. ∫duu=ln|u|+C
3. ∫eudu=eu+C
4. ∫audu=aulna+C
5. ∫sinudu=−cosu+C
6. ∫cosudu=sinu+C
7. ∫sec2udu=tanu+C
8. ∫csc2udu=−cotu+C
9. ∫secutanudu=secu+C
10. ∫cscucotudu=−cscu+C
11. ∫tanudu=ln|secu|+C
12. ∫cotudu=ln|sinu|+C
13. ∫secudu=ln|secu+tanu|+C
14. ∫cscudu=ln|cscu−cotu|+C
15. ∫du√a2−u2=sin−1(ua)+C
16. ∫dua2+u2=1atan−1(ua)+C
17. ∫duu√u2−a2=1asec−1|u|a+C
Integrais trigonométricas
18. ∫sin2udu=12u−14sin2u+C
19. ∫cos2udu=12u+14sin2u+C
20. ∫tan2udu=tanu−u+C
21. ∫cot2udu=−cotu−u+C
22. ∫sin3udu=−13(2+sin2u)cosu+C
23. ∫cos3udu=13(2+cos2u)sinu+C
24. ∫tan3udu=12tan2u+ln|cosu|+C
25. ∫cot3udu=−12cot2u−ln|sinu|+C
26. ∫sec3udu=12secutanu+12ln|secu+tanu|+C
27. ∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C
28. ∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu
29. ∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu
30. ∫tannudu=1n−1tann−1u−∫tann−2udu
31. ∫cotnudu=−1n−1cotn−1u−∫cotn−2udu
32. ∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu
33. ∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu
34. ∫sinausinbudu=sin(a−b)u2(a−b)−sin(a+b)u2(a+b)+C
35. ∫cosaucosbudu=sin(a−b)u2(a−b)+sin(a+b)u2(a+b)+C
36. ∫sinaucosbudu=−cos(a−b)u2(a−b)−cos(a+b)u2(a+b)+C
37. ∫usinudu=sinu−ucosu+C
38. ∫ucosudu=cosu+usinu+C
39. ∫unsinudu=−uncosu+n∫un−1cosudu
40. ∫uncosudu=unsinu−n∫un−1sinudu
41. ∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu
Integrais exponenciais e logarítmicas
42. ∫ueaudu=1a2(au−1)eau+C
43. ∫uneaudu=1auneau−na∫un−1eaudu
44. ∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C
45. ∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C
46. ∫lnudu=ulnu−u+C
47. ∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C
48. ∫1ulnudu=ln|lnu|+C
Integrais hiperbólicas
49. ∫sinhudu=coshu+C
50. ∫coshudu=sinhu+C
51. ∫tanhudu=lncoshu+C
52. ∫cothudu=ln|sinhu|+C
53. ∫sechudu=tan−1|sinhu|+C
54. ∫cschudu=ln∣tanh12u∣+C
55. ∫sech2udu=tanhu+C
56. ∫csch2udu=−cothu+C
57. ∫sechutanhudu=−sechu+C
58. ∫cschucothudu=−cschu+C
Integrais trigonométricas inversas
59. ∫sin−1udu=usin−1u+√1−u2+C
60. ∫cos−1udu=ucos−1u−√1−u2+C
61. ∫tan−1udu=utan−1u−12ln(1+u2)+C
62. ∫usin−1udu=2u2−14sin−1u+u√1−u24+C
63. ∫ucos−1udu=2u2−14cos−1u−u√1−u24+C
64. ∫utan−1udu=u2+12tan−1u−u2+C
65. ∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du√1−u2],n≠−1
66. ∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du√1−u2],n≠−1
67. ∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1
Integrais envolvendo um 2 + u 2, a > 0
68. ∫√a2+u2du=u2√a2+u2+a22ln(u+√a2+u2)+C
69. ∫u2√a2+u2du=u8(a2+2u2)√a2+u2−a48ln(u+√a2+u2)+C
70. ∫√a2+u2udu=√a2+u2−aln|a+√a2+u2u|+C
71. ∫√a2+u2u2du=−√a2+u2u+ln(u+√a2+u2)+C
72. ∫du√a2+u2=ln(u+√a2+u2)+C
73. ∫u2√a2+u2du=u2(√a2+u2)−a22ln(u+√a2+u2)+C
74. ∫duu√a2+u2=−1aln|√a2+u2+au|+C
75. ∫duu2√a2+u2=−√a2+u2a2u+C
76. ∫du(a2+u2)3/2=ua2√a2+u2+C
Integrais envolvendo u 2 − a 2, a > 0
77. ∫√u2−a2du=u2√u2−a2−a22ln|u+√u2−a2|+C
78. ∫u2√u2−a2du=u8(2u2−a2)√u2−a2−a48ln|u+√u2−a2|+C
79. ∫√u2−a2udu=√u2−a2−acos−1a|u|+C
80. ∫√u2−a2u2du=−√u2−a2u+ln|u+√u2−a2|+C
81. ∫du√u2−a2=ln|u+√u2−a2|+C
82. ∫u2√u2−a2du=u2√u2−a2+a22ln|u+√u2−a2|+C
83. ∫duu2√u2−a2=√u2−a2a2u+C
84. ∫du(u2−a2)3/2=−ua2√u2−a2+C
Integrais envolvendo um 2 − u 2, a > 0
85. ∫√a2−u2du=u2√a2−u2+a22sin−1ua+C
86. ∫u2√a2−u2du=u8(2u2−a2)√a2−u2+a48sin−1ua+C
87. ∫√a2−u2udu=√a2−u2−aln|a+√a2−u2u|+C
88. ∫√a2−u2u2du=−1u√a2−u2−sin−1ua+C
89. ∫u2√a2−u2du=12(−u√a2−u2+a2sin−1ua)+C
90. ∫duu√a2−u2=−1aln|a+√a2−u2u|+C
91. ∫duu2√a2−u2=−1a2u√a2−u2+C
92. ∫(a2−u2)3/2du=−u8(2u2−5a2)√a2−u2+3a48sin−1ua+C
93. ∫du(a2−u2)3/2=−ua2√a2−u2+C
Integrais envolvendo 2 au − u 2, a > 0
94. ∫√2au−u2du=u−a2√2au−u2+a22cos−1(a−ua)+C
95. ∫du√2au−u2=cos−1(a−ua)+C
96. ∫u√2au−u2du=2u2−au−3a26√2au−u2+a32cos−1(a−ua)+C
97. ∫duu√2au−u2=−√2au−u2au+C
Integrais envolvendo um + bu, a ≠ 0
98. ∫ua+budu=1b2(a+bu−aln|a+bu|)+C
99. ∫u2a+budu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C
100. ∫duu(a+bu)=1aln|ua+bu|+C
101. ∫duu2(a+bu)=−1au+ba2ln|a+buu|+C
102. ∫u(a+bu)2du=ab2(a+bu)+1b2ln|a+bu|+C
103. ∫uu(a+bu)2du=1a(a+bu)−1a2ln|a+buu|+C
104. ∫u2(a+bu)2du=1b3(a+bu−a2a+bu−2aln|a+bu|)+C
105. ∫u√a+budu=215b2(3bu−2a)(a+bu)3/2+C
106. ∫u√a+budu=23b2(bu−2a)√a+bu+C
107. ∫u2√a+budu=215b3(8a2+3b2u2−4abu)√a+bu+C
108. ∫duu√a+bu={1√aln|√a+bu−√a√a+bu+√a|+C,ifa>0√2√−atan−1√a+bu−a+C,ifa<0
109. ∫√a+buudu=2√a+bu+a∫duu√a+bu
110. ∫√a+buu2du=−√a+buu+b2∫duu√a+bu
111. ∫un√a+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1√a+budu]
112. ∫un√a+budu=2un√a+bub(2n+1)−2nab(2n+1)∫un−1√a+budu
113. ∫duun√a+bu=−√a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1√a+bu