Calculus Volume 2

# A | Table of Integrals

Calculus Volume 2A | Table of Integrals

## Basic Integrals

1. $∫undu=un+1n+1+C,n≠−1∫undu=un+1n+1+C,n≠−1$

2. $∫duu=ln|u|+C∫duu=ln|u|+C$

3. $∫eudu=eu+C∫eudu=eu+C$

4. $∫audu=aulna+C∫audu=aulna+C$

5. $∫sinudu=−cosu+C∫sinudu=−cosu+C$

6. $∫cosudu=sinu+C∫cosudu=sinu+C$

7. $∫sec2udu=tanu+C∫sec2udu=tanu+C$

8. $∫csc2udu=−cotu+C∫csc2udu=−cotu+C$

9. $∫secutanudu=secu+C∫secutanudu=secu+C$

10. $∫cscucotudu=−cscu+C∫cscucotudu=−cscu+C$

11. $∫tanudu=ln|secu|+C∫tanudu=ln|secu|+C$

12. $∫cotudu=ln|sinu|+C∫cotudu=ln|sinu|+C$

13. $∫secudu=ln|secu+tanu|+C∫secudu=ln|secu+tanu|+C$

14. $∫cscudu=ln|cscu−cotu|+C∫cscudu=ln|cscu−cotu|+C$

15. $∫dua2−u2=sin−1ua+C∫dua2−u2=sin−1ua+C$

16. $∫dua2+u2=1atan−1ua+C∫dua2+u2=1atan−1ua+C$

17. $∫duuu2−a2=1asec−1ua+C∫duuu2−a2=1asec−1ua+C$

## Trigonometric Integrals

18. $∫sin2udu=12u−14sin2u+C∫sin2udu=12u−14sin2u+C$

19. $∫cos2udu=12u+14sin2u+C∫cos2udu=12u+14sin2u+C$

20. $∫tan2udu=tanu−u+C∫tan2udu=tanu−u+C$

21. $∫cot2udu=−cotu−u+C∫cot2udu=−cotu−u+C$

22. $∫sin3udu=−13(2+sin2u)cosu+C∫sin3udu=−13(2+sin2u)cosu+C$

23. $∫cos3udu=13(2+cos2u)sinu+C∫cos3udu=13(2+cos2u)sinu+C$

24. $∫tan3udu=12tan2u+ln|cosu|+C∫tan3udu=12tan2u+ln|cosu|+C$

25. $∫cot3udu=−12cot2u−ln|sinu|+C∫cot3udu=−12cot2u−ln|sinu|+C$

26. $∫sec3udu=12secutanu+12ln|secu+tanu|+C∫sec3udu=12secutanu+12ln|secu+tanu|+C$

27. $∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C$

28. $∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu$

29. $∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu$

30. $∫tannudu=1n−1tann−1u−∫tann−2udu∫tannudu=1n−1tann−1u−∫tann−2udu$

31. $∫cotnudu=−1n−1cotn−1u−∫cotn−2udu∫cotnudu=−1n−1cotn−1u−∫cotn−2udu$

32. $∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu$

33. $∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu$

34. $∫sinausinbudu=sin(a−b)u2(a−b)−sin(a+b)u2(a+b)+C∫sinausinbudu=sin(a−b)u2(a−b)−sin(a+b)u2(a+b)+C$

35. $∫cosaucosbudu=sin(a−b)u2(a−b)+sin(a+b)u2(a+b)+C∫cosaucosbudu=sin(a−b)u2(a−b)+sin(a+b)u2(a+b)+C$

36. $∫sinaucosbudu=−cos(a−b)u2(a−b)−cos(a+b)u2(a+b)+C∫sinaucosbudu=−cos(a−b)u2(a−b)−cos(a+b)u2(a+b)+C$

37. $∫usinudu=sinu−ucosu+C∫usinudu=sinu−ucosu+C$

38. $∫ucosudu=cosu+usinu+C∫ucosudu=cosu+usinu+C$

39. $∫unsinudu=−uncosu+n∫un−1cosudu∫unsinudu=−uncosu+n∫un−1cosudu$

40. $∫uncosudu=unsinu−n∫un−1sinudu∫uncosudu=unsinu−n∫un−1sinudu$

41. $∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu$

## Exponential and Logarithmic Integrals

42. $∫ueaudu=1a2(au−1)eau+C∫ueaudu=1a2(au−1)eau+C$

43. $∫uneaudu=1auneau−na∫un−1eaudu∫uneaudu=1auneau−na∫un−1eaudu$

44. $∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C$

45. $∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C$

46. $∫lnudu=ulnu−u+C∫lnudu=ulnu−u+C$

47. $∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C$

48. $∫1ulnudu=ln|lnu|+C∫1ulnudu=ln|lnu|+C$

## Hyperbolic Integrals

49. $∫sinhudu=coshu+C∫sinhudu=coshu+C$

50. $∫coshudu=sinhu+C∫coshudu=sinhu+C$

51. $∫tanhudu=lncoshu+C∫tanhudu=lncoshu+C$

52. $∫cothudu=ln|sinhu|+C∫cothudu=ln|sinhu|+C$

53. $∫sechudu=tan−1|sinhu|+C∫sechudu=tan−1|sinhu|+C$

54. $∫cschudu=ln|tanh12u|+C∫cschudu=ln|tanh12u|+C$

55. $∫sech2udu=tanhu+C∫sech2udu=tanhu+C$

56. $∫csch2udu=−cothu+C∫csch2udu=−cothu+C$

57. $∫sechutanhudu=−sechu+C∫sechutanhudu=−sechu+C$

58. $∫cschucothudu=−cschu+C∫cschucothudu=−cschu+C$

## Inverse Trigonometric Integrals

59. $∫sin−1udu=usin−1u+1−u2+C∫sin−1udu=usin−1u+1−u2+C$

60. $∫cos−1udu=ucos−1u−1−u2+C∫cos−1udu=ucos−1u−1−u2+C$

61. $∫tan−1udu=utan−1u−12ln(1+u2)+C∫tan−1udu=utan−1u−12ln(1+u2)+C$

62. $∫usin−1udu=2u2−14sin−1u+u1−u24+C∫usin−1udu=2u2−14sin−1u+u1−u24+C$

63. $∫ucos−1udu=2u2−14cos−1u−u1−u24+C∫ucos−1udu=2u2−14cos−1u−u1−u24+C$

64. $∫utan−1udu=u2+12tan−1u−u2+C∫utan−1udu=u2+12tan−1u−u2+C$

65. $∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du1−u2],n≠−1∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du1−u2],n≠−1$

66. $∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du1−u2],n≠−1∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du1−u2],n≠−1$

67. $∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1$

## Integrals Involving a2 + u2, a > 0

68. $∫a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C∫a2+u2du=u2a2+u2+a22ln(u+a2+u2)+C$

69. $∫u2a2+u2du=u8(a2+2u2)a2+u2−a48ln(u+a2+u2)+C∫u2a2+u2du=u8(a2+2u2)a2+u2−a48ln(u+a2+u2)+C$

70. $∫a2+u2udu=a2+u2−aln|a+a2+u2u|+C∫a2+u2udu=a2+u2−aln|a+a2+u2u|+C$

71. $∫a2+u2u2du=−a2+u2u+ln(u+a2+u2)+C∫a2+u2u2du=−a2+u2u+ln(u+a2+u2)+C$

72. $∫dua2+u2=ln(u+a2+u2)+C∫dua2+u2=ln(u+a2+u2)+C$

73. $∫u2dua2+u2=u2(a2+u2)−a22ln(u+a2+u2)+C∫u2dua2+u2=u2(a2+u2)−a22ln(u+a2+u2)+C$

74. $∫duua2+u2=−1aln|a2+u2+au|+C∫duua2+u2=−1aln|a2+u2+au|+C$

75. $∫duu2a2+u2=−a2+u2a2u+C∫duu2a2+u2=−a2+u2a2u+C$

76. $∫du(a2+u2)3/2=ua2a2+u2+C∫du(a2+u2)3/2=ua2a2+u2+C$

## Integrals Involving u2 − a2, a > 0

77. $∫u2−a2du=u2u2−a2−a22ln|u+u2−a2|+C∫u2−a2du=u2u2−a2−a22ln|u+u2−a2|+C$

78. $∫u2u2−a2du=u8(2u2−a2)u2−a2−a48ln|u+u2−a2|+C∫u2u2−a2du=u8(2u2−a2)u2−a2−a48ln|u+u2−a2|+C$

79. $∫u2−a2udu=u2−a2−acos−1a|u|+C∫u2−a2udu=u2−a2−acos−1a|u|+C$

80. $∫u2−a2u2du=−u2−a2u+ln|u+u2−a2|+C∫u2−a2u2du=−u2−a2u+ln|u+u2−a2|+C$

81. $∫duu2−a2=ln|u+u2−a2|+C∫duu2−a2=ln|u+u2−a2|+C$

82. $∫u2duu2−a2=u2u2−a2+a22ln|u+u2−a2|+C∫u2duu2−a2=u2u2−a2+a22ln|u+u2−a2|+C$

83. $∫duu2u2−a2=u2−a2a2u+C∫duu2u2−a2=u2−a2a2u+C$

84a. $∫du(u2−a2)3/2=−ua2u2−a2+C∫du(u2−a2)3/2=−ua2u2−a2+C$

84b. $∫duu2-a2=12alnu-au+a+C∫duu2-a2=12alnu-au+a+C$

## Integrals Involving a2 − u2, a > 0

85. $∫a2−u2du=u2a2−u2+a22sin−1ua+C∫a2−u2du=u2a2−u2+a22sin−1ua+C$

86. $∫u2a2−u2du=u8(2u2−a2)a2−u2+a48sin−1ua+C∫u2a2−u2du=u8(2u2−a2)a2−u2+a48sin−1ua+C$

87. $∫a2−u2udu=a2−u2−aln|a+a2−u2u|+C∫a2−u2udu=a2−u2−aln|a+a2−u2u|+C$

88. $∫a2−u2u2du=−1ua2−u2−sin−1ua+C∫a2−u2u2du=−1ua2−u2−sin−1ua+C$

89. $∫u2dua2−u2=−u2a2−u2+a22sin−1ua+C∫u2dua2−u2=−u2a2−u2+a22sin−1ua+C$

90. $∫duua2−u2=−1aln|a+a2−u2u|+C∫duua2−u2=−1aln|a+a2−u2u|+C$

91. $∫duu2a2−u2=−1a2ua2−u2+C∫duu2a2−u2=−1a2ua2−u2+C$

92. $∫(a2−u2)3/2du=−u8(2u2−5a2)a2−u2+3a48sin−1ua+C∫(a2−u2)3/2du=−u8(2u2−5a2)a2−u2+3a48sin−1ua+C$

93a. $∫du(a2−u2)3/2=ua2a2−u2+C∫du(a2−u2)3/2=ua2a2−u2+C$

93b. $∫dua2-u2=12alnu+au-a+C∫dua2-u2=12alnu+au-a+C$

## Integrals Involving 2au − u2, a > 0

94. $∫2au−u2du=u−a22au−u2+a22cos−1(a−ua)+C∫2au−u2du=u−a22au−u2+a22cos−1(a−ua)+C$

95. $∫du2au−u2=cos−1(a−ua)+C∫du2au−u2=cos−1(a−ua)+C$

96. $∫u2au−u2du=2u2−au−3a262au−u2+a32cos−1(a−ua)+C∫u2au−u2du=2u2−au−3a262au−u2+a32cos−1(a−ua)+C$

97. $∫duu2au−u2=−2au−u2au+C∫duu2au−u2=−2au−u2au+C$

## Integrals Involving a + bu, a ≠ 0

98. $∫udua+bu=1b2(a+bu−aln|a+bu|)+C∫udua+bu=1b2(a+bu−aln|a+bu|)+C$

99. $∫u2dua+bu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C∫u2dua+bu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C$

100. $∫duu(a+bu)=1aln|ua+bu|+C∫duu(a+bu)=1aln|ua+bu|+C$

101. $∫duu2(a+bu)=−1au+ba2ln|a+buu|+C∫duu2(a+bu)=−1au+ba2ln|a+buu|+C$

102. $∫udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+C∫udu(a+bu)2=ab2(a+bu)+1b2ln|a+bu|+C$

103. $∫uduu(a+bu)2=1a(a+bu)−1a2ln|a+buu|+C∫uduu(a+bu)2=1a(a+bu)−1a2ln|a+buu|+C$

104. $∫u2du(a+bu)2=1b3(a+bu−a2a+bu−2aln|a+bu|)+C∫u2du(a+bu)2=1b3(a+bu−a2a+bu−2aln|a+bu|)+C$

105. $∫ua+budu=215b2(3bu−2a)(a+bu)3/2+C∫ua+budu=215b2(3bu−2a)(a+bu)3/2+C$

106. $∫udua+bu=23b2(bu−2a)a+bu+C∫udua+bu=23b2(bu−2a)a+bu+C$

107. $∫u2dua+bu=215b3(8a2+3b2u2−4abu)a+bu+C∫u2dua+bu=215b3(8a2+3b2u2−4abu)a+bu+C$

108. $∫duua+bu=1aln|a+bu−aa+bu+a|+C,ifa>0=2−atan−1a+bu−a+C,ifa<0∫duua+bu=1aln|a+bu−aa+bu+a|+C,ifa>0=2−atan−1a+bu−a+C,ifa<0$

109. $∫a+buudu=2a+bu+a∫duua+bu∫a+buudu=2a+bu+a∫duua+bu$

110. $∫a+buu2du=−a+buu+b2∫duua+bu∫a+buu2du=−a+buu+b2∫duua+bu$

111. $∫una+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1a+budu]∫una+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1a+budu]$

112. $∫undua+bu=2una+bub(2n+1)−2nab(2n+1)∫un−1dua+bu∫undua+bu=2una+bub(2n+1)−2nab(2n+1)∫un−1dua+bu$

113. $∫duuna+bu=−a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1a+bu∫duuna+bu=−a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1a+bu$

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