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Calculus Volume 3

A | Table of Integrals

Calculus Volume 3A | Table of Integrals

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Table of contents
  1. Preface
  2. 1 Parametric Equations and Polar Coordinates
    1. Introduction
    2. 1.1 Parametric Equations
    3. 1.2 Calculus of Parametric Curves
    4. 1.3 Polar Coordinates
    5. 1.4 Area and Arc Length in Polar Coordinates
    6. 1.5 Conic Sections
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Vectors in Space
    1. Introduction
    2. 2.1 Vectors in the Plane
    3. 2.2 Vectors in Three Dimensions
    4. 2.3 The Dot Product
    5. 2.4 The Cross Product
    6. 2.5 Equations of Lines and Planes in Space
    7. 2.6 Quadric Surfaces
    8. 2.7 Cylindrical and Spherical Coordinates
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Vector-Valued Functions
    1. Introduction
    2. 3.1 Vector-Valued Functions and Space Curves
    3. 3.2 Calculus of Vector-Valued Functions
    4. 3.3 Arc Length and Curvature
    5. 3.4 Motion in Space
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Differentiation of Functions of Several Variables
    1. Introduction
    2. 4.1 Functions of Several Variables
    3. 4.2 Limits and Continuity
    4. 4.3 Partial Derivatives
    5. 4.4 Tangent Planes and Linear Approximations
    6. 4.5 The Chain Rule
    7. 4.6 Directional Derivatives and the Gradient
    8. 4.7 Maxima/Minima Problems
    9. 4.8 Lagrange Multipliers
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Multiple Integration
    1. Introduction
    2. 5.1 Double Integrals over Rectangular Regions
    3. 5.2 Double Integrals over General Regions
    4. 5.3 Double Integrals in Polar Coordinates
    5. 5.4 Triple Integrals
    6. 5.5 Triple Integrals in Cylindrical and Spherical Coordinates
    7. 5.6 Calculating Centers of Mass and Moments of Inertia
    8. 5.7 Change of Variables in Multiple Integrals
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Vector Calculus
    1. Introduction
    2. 6.1 Vector Fields
    3. 6.2 Line Integrals
    4. 6.3 Conservative Vector Fields
    5. 6.4 Green’s Theorem
    6. 6.5 Divergence and Curl
    7. 6.6 Surface Integrals
    8. 6.7 Stokes’ Theorem
    9. 6.8 The Divergence Theorem
    10. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. 7 Second-Order Differential Equations
    1. Introduction
    2. 7.1 Second-Order Linear Equations
    3. 7.2 Nonhomogeneous Linear Equations
    4. 7.3 Applications
    5. 7.4 Series Solutions of Differential Equations
    6. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  9. A | Table of Integrals
  10. B | Table of Derivatives
  11. C | Review of Pre-Calculus
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
  13. Index

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)u2(ab)sin(a+b)u2(a+b)+Csinausinbudu=sin(ab)u2(ab)sin(a+b)u2(a+b)+C

35. cosaucosbudu=sin(ab)u2(ab)+sin(a+b)u2(a+b)+Ccosaucosbudu=sin(ab)u2(ab)+sin(a+b)u2(a+b)+C

36. sinaucosbudu=cos(ab)u2(ab)cos(a+b)u2(a+b)+Csinaucosbudu=cos(ab)u2(ab)cos(a+b)u2(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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