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Calculus Volume 2

A | Table of Integrals

Calculus Volume 2A | Table of Integrals

Basic Integrals

1. undu=un+1n+1+C,n1undu=un+1n+1+C,n1

2. duu=ln|u|+Cduu=ln|u|+C

3. eudu=eu+Ceudu=eu+C

4. audu=aulna+Caudu=aulna+C

5. sinudu=−cosu+Csinudu=−cosu+C

6. cosudu=sinu+Ccosudu=sinu+C

7. sec2udu=tanu+Csec2udu=tanu+C

8. csc2udu=−cotu+Ccsc2udu=−cotu+C

9. secutanudu=secu+Csecutanudu=secu+C

10. cscucotudu=−cscu+Ccscucotudu=−cscu+C

11. tanudu=ln|secu|+Ctanudu=ln|secu|+C

12. cotudu=ln|sinu|+Ccotudu=ln|sinu|+C

13. secudu=ln|secu+tanu|+Csecudu=ln|secu+tanu|+C

14. cscudu=ln|cscucotu|+Ccscudu=ln|cscucotu|+C

15. dua2u2=sin−1ua+Cdua2u2=sin−1ua+C

16. dua2+u2=1atan−1ua+Cdua2+u2=1atan−1ua+C

17. duuu2a2=1asec−1ua+Cduuu2a2=1asec−1ua+C

Trigonometric Integrals

18. sin2udu=12u14sin2u+Csin2udu=12u14sin2u+C

19. cos2udu=12u+14sin2u+Ccos2udu=12u+14sin2u+C

20. tan2udu=tanuu+Ctan2udu=tanuu+C

21. cot2udu=cotuu+Ccot2udu=cotuu+C

22. sin3udu=13(2+sin2u)cosu+Csin3udu=13(2+sin2u)cosu+C

23. cos3udu=13(2+cos2u)sinu+Ccos3udu=13(2+cos2u)sinu+C

24. tan3udu=12tan2u+ln|cosu|+Ctan3udu=12tan2u+ln|cosu|+C

25. cot3udu=12cot2uln|sinu|+Ccot3udu=12cot2uln|sinu|+C

26. sec3udu=12secutanu+12ln|secu+tanu|+Csec3udu=12secutanu+12ln|secu+tanu|+C

27. csc3udu=12cscucotu+12ln|cscucotu|+Ccsc3udu=12cscucotu+12ln|cscucotu|+C

28. sinnudu=1nsinn1ucosu+n1nsinn2udusinnudu=1nsinn1ucosu+n1nsinn2udu

29. cosnudu=1ncosn1usinu+n1ncosn2uducosnudu=1ncosn1usinu+n1ncosn2udu

30. tannudu=1n1tann1utann2udutannudu=1n1tann1utann2udu

31. cotnudu=−1n1cotn1ucotn2uducotnudu=−1n1cotn1ucotn2udu

32. secnudu=1n1tanusecn2u+n2n1secn2udusecnudu=1n1tanusecn2u+n2n1secn2udu

33. cscnudu=−1n1cotucscn2u+n2n1cscn2uducscnudu=−1n1cotucscn2u+n2n1cscn2udu

34. sinausinbudu=sin((ab)u)2(ab)sin((a+b)u)2(a+b)+Csinausinbudu=sin((ab)u)2(ab)sin((a+b)u)2(a+b)+C

35. cosaucosbudu=sin((ab)u)2(ab)+sin((a+b)u)2(a+b)+Ccosaucosbudu=sin((ab)u)2(ab)+sin((a+b)u)2(a+b)+C

36. sinaucosbudu=cos((ab)u)2(ab)cos((a+b)u)2(a+b)+Csinaucosbudu=cos((ab)u)2(ab)cos((a+b)u)2(a+b)+C

37. usinudu=sinuucosu+Cusinudu=sinuucosu+C

38. ucosudu=cosu+usinu+Cucosudu=cosu+usinu+C

39. unsinudu=uncosu+nun1cosuduunsinudu=uncosu+nun1cosudu

40. uncosudu=unsinunun1sinuduuncosudu=unsinunun1sinudu

41. sinnucosmudu=sinn1ucosm+1un+m+n1n+msinn2ucosmudu=sinn+1ucosm1un+m+m1n+msinnucosm2udusinnucosmudu=sinn1ucosm+1un+m+n1n+msinn2ucosmudu=sinn+1ucosm1un+m+m1n+msinnucosm2udu

Exponential and Logarithmic Integrals

42. ueaudu=1a2(au1)eau+Cueaudu=1a2(au1)eau+C

43. uneaudu=1auneaunaun1eauduuneaudu=1auneaunaun1eaudu

44. eausinbudu=eaua2+b2(asinbubcosbu)+Ceausinbudu=eaua2+b2(asinbubcosbu)+C

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

46. lnudu=ulnuu+Clnudu=ulnuu+C

47. unlnudu=un+1(n+1)2[(n+1)lnu1]+Cunlnudu=un+1(n+1)2[(n+1)lnu1]+C

48. 1ulnudu=ln|lnu|+C1ulnudu=ln|lnu|+C

Hyperbolic Integrals

49. sinhudu=coshu+Csinhudu=coshu+C

50. coshudu=sinhu+Ccoshudu=sinhu+C

51. tanhudu=lncoshu+Ctanhudu=lncoshu+C

52. cothudu=ln|sinhu|+Ccothudu=ln|sinhu|+C

53. sechudu=tan−1|sinhu|+Csechudu=tan−1|sinhu|+C

54. cschudu=ln|tanh12u|+Ccschudu=ln|tanh12u|+C

55. sech2udu=tanhu+Csech2udu=tanhu+C

56. csch2udu=cothu+Ccsch2udu=cothu+C

57. sechutanhudu=sechu+Csechutanhudu=sechu+C

58. cschucothudu=cschu+Ccschucothudu=cschu+C

Inverse Trigonometric Integrals

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

60. cos−1udu=ucos−1u1u2+Ccos−1udu=ucos−1u1u2+C

61. tan−1udu=utan−1u12ln(1+u2)+Ctan−1udu=utan−1u12ln(1+u2)+C

62. usin−1udu=2u214sin−1u+u1u24+Cusin−1udu=2u214sin−1u+u1u24+C

63. ucos−1udu=2u214cos−1uu1u24+Cucos−1udu=2u214cos−1uu1u24+C

64. utan−1udu=u2+12tan−1uu2+Cutan−1udu=u2+12tan−1uu2+C

65. unsin−1udu=1n+1[un+1sin−1uun+1du1u2],n1unsin−1udu=1n+1[un+1sin−1uun+1du1u2],n1

66. uncos−1udu=1n+1[un+1cos−1u+un+1du1u2],n1uncos−1udu=1n+1[un+1cos−1u+un+1du1u2],n1

67. untan−1udu=1n+1[un+1tan−1uun+1du1+u2],n1untan−1udu=1n+1[un+1tan−1uun+1du1+u2],n1

Integrals Involving a2 + u2, a > 0

68. a2+u2du=u2a2+u2+a22ln(u+a2+u2)+Ca2+u2du=u2a2+u2+a22ln(u+a2+u2)+C

69. u2a2+u2du=u8(a2+2u2)a2+u2a48ln(u+a2+u2)+Cu2a2+u2du=u8(a2+2u2)a2+u2a48ln(u+a2+u2)+C

70. a2+u2udu=a2+u2aln|a+a2+u2u|+Ca2+u2udu=a2+u2aln|a+a2+u2u|+C

71. a2+u2u2du=a2+u2u+ln(u+a2+u2)+Ca2+u2u2du=a2+u2u+ln(u+a2+u2)+C

72. dua2+u2=ln(u+a2+u2)+Cdua2+u2=ln(u+a2+u2)+C

73. u2dua2+u2=u2(a2+u2)a22ln(u+a2+u2)+Cu2dua2+u2=u2(a2+u2)a22ln(u+a2+u2)+C

74. duua2+u2=1aln|a2+u2+au|+Cduua2+u2=1aln|a2+u2+au|+C

75. duu2a2+u2=a2+u2a2u+Cduu2a2+u2=a2+u2a2u+C

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

Integrals Involving u2a2, a > 0

77. u2a2du=u2u2a2a22ln|u+u2a2|+Cu2a2du=u2u2a2a22ln|u+u2a2|+C

78. u2u2a2du=u8(2u2a2)u2a2a48ln|u+u2a2|+Cu2u2a2du=u8(2u2a2)u2a2a48ln|u+u2a2|+C

79. u2a2udu=u2a2acos−1a|u|+Cu2a2udu=u2a2acos−1a|u|+C

80. u2a2u2du=u2a2u+ln|u+u2a2|+Cu2a2u2du=u2a2u+ln|u+u2a2|+C

81. duu2a2=ln|u+u2a2|+Cduu2a2=ln|u+u2a2|+C

82. u2duu2a2=u2u2a2+a22ln|u+u2a2|+Cu2duu2a2=u2u2a2+a22ln|u+u2a2|+C

83. duu2u2a2=u2a2a2u+Cduu2u2a2=u2a2a2u+C

84a. du(u2a2)3/2=ua2u2a2+Cdu(u2a2)3/2=ua2u2a2+C

84b. duu2-a2=12alnu-au+a+Cduu2-a2=12alnu-au+a+C

Integrals Involving a2u2, a > 0

85. a2u2du=u2a2u2+a22sin−1ua+Ca2u2du=u2a2u2+a22sin−1ua+C

86. u2a2u2du=u8(2u2a2)a2u2+a48sin−1ua+Cu2a2u2du=u8(2u2a2)a2u2+a48sin−1ua+C

87. a2u2udu=a2u2aln|a+a2u2u|+Ca2u2udu=a2u2aln|a+a2u2u|+C

88. a2u2u2du=1ua2u2sin−1ua+Ca2u2u2du=1ua2u2sin−1ua+C

89. u2dua2u2=u2a2u2+a22sin−1ua+Cu2dua2u2=u2a2u2+a22sin−1ua+C

90. duua2u2=1aln|a+a2u2u|+Cduua2u2=1aln|a+a2u2u|+C

91. duu2a2u2=1a2ua2u2+Cduu2a2u2=1a2ua2u2+C

92. (a2u2)3/2du=u8(2u25a2)a2u2+3a48sin−1ua+C(a2u2)3/2du=u8(2u25a2)a2u2+3a48sin−1ua+C

93a. du(a2u2)3/2=ua2a2u2+Cdu(a2u2)3/2=ua2a2u2+C

93b. dua2-u2=12alnu+au-a+Cdua2-u2=12alnu+au-a+C

Integrals Involving 2auu2, a > 0

94. 2auu2du=ua22auu2+a22cos−1(aua)+C2auu2du=ua22auu2+a22cos−1(aua)+C

95. du2auu2=cos−1(aua)+Cdu2auu2=cos−1(aua)+C

96. u2auu2du=2u2au3a262auu2+a32cos−1(aua)+Cu2auu2du=2u2au3a262auu2+a32cos−1(aua)+C

97. duu2auu2=2auu2au+Cduu2auu2=2auu2au+C

Integrals Involving a + bu, a ≠ 0

98. udua+bu=1b2(a+bualn|a+bu|)+Cudua+bu=1b2(a+bualn|a+bu|)+C

99. u2dua+bu=12b3[(a+bu)24a(a+bu)+2a2ln|a+bu|]+Cu2dua+bu=12b3[(a+bu)24a(a+bu)+2a2ln|a+bu|]+C

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

101. duu2(a+bu)=1au+ba2ln|a+buu|+Cduu2(a+bu)=1au+ba2ln|a+buu|+C

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

103. uduu(a+bu)2=1a(a+bu)1a2ln|a+buu|+Cuduu(a+bu)2=1a(a+bu)1a2ln|a+buu|+C

104. u2du(a+bu)2=1b3(a+bua2a+bu2aln|a+bu|)+Cu2du(a+bu)2=1b3(a+bua2a+bu2aln|a+bu|)+C

105. ua+budu=215b2(3bu2a)(a+bu)3/2+Cua+budu=215b2(3bu2a)(a+bu)3/2+C

106. udua+bu=23b2(bu2a)a+bu+Cudua+bu=23b2(bu2a)a+bu+C

107. u2dua+bu=215b3(8a2+3b2u24abu)a+bu+Cu2dua+bu=215b3(8a2+3b2u24abu)a+bu+C

108. duua+bu=1aln|a+buaa+bu+a|+C,ifa>0=2atan1a+bua+C,ifa<0duua+bu=1aln|a+buaa+bu+a|+C,ifa>0=2atan1a+bua+C,ifa<0

109. a+buudu=2a+bu+aduua+bua+buudu=2a+bu+aduua+bu

110. a+buu2du=a+buu+b2duua+bua+buu2du=a+buu+b2duua+bu

111. una+budu=2b(2n+3)[un(a+bu)3/2naun1a+budu]una+budu=2b(2n+3)[un(a+bu)3/2naun1a+budu]

112. undua+bu=2una+bub(2n+1)2nab(2n+1)un1dua+buundua+bu=2una+bub(2n+1)2nab(2n+1)un1dua+bu

113. duuna+bu=a+bua(n1)un1b(2n3)2a(n1)duun1a+buduuna+bu=a+bua(n1)un1b(2n3)2a(n1)duun1a+bu

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