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Calculus Volume 1

A | Table of Integrals

Calculus Volume 1A | Table of Integrals

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Table of contents
  1. Preface
  2. 1 Functions and Graphs
    1. Introduction
    2. 1.1 Review of Functions
    3. 1.2 Basic Classes of Functions
    4. 1.3 Trigonometric Functions
    5. 1.4 Inverse Functions
    6. 1.5 Exponential and Logarithmic Functions
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  3. 2 Limits
    1. Introduction
    2. 2.1 A Preview of Calculus
    3. 2.2 The Limit of a Function
    4. 2.3 The Limit Laws
    5. 2.4 Continuity
    6. 2.5 The Precise Definition of a Limit
    7. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  4. 3 Derivatives
    1. Introduction
    2. 3.1 Defining the Derivative
    3. 3.2 The Derivative as a Function
    4. 3.3 Differentiation Rules
    5. 3.4 Derivatives as Rates of Change
    6. 3.5 Derivatives of Trigonometric Functions
    7. 3.6 The Chain Rule
    8. 3.7 Derivatives of Inverse Functions
    9. 3.8 Implicit Differentiation
    10. 3.9 Derivatives of Exponential and Logarithmic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  5. 4 Applications of Derivatives
    1. Introduction
    2. 4.1 Related Rates
    3. 4.2 Linear Approximations and Differentials
    4. 4.3 Maxima and Minima
    5. 4.4 The Mean Value Theorem
    6. 4.5 Derivatives and the Shape of a Graph
    7. 4.6 Limits at Infinity and Asymptotes
    8. 4.7 Applied Optimization Problems
    9. 4.8 L’Hôpital’s Rule
    10. 4.9 Newton’s Method
    11. 4.10 Antiderivatives
    12. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  6. 5 Integration
    1. Introduction
    2. 5.1 Approximating Areas
    3. 5.2 The Definite Integral
    4. 5.3 The Fundamental Theorem of Calculus
    5. 5.4 Integration Formulas and the Net Change Theorem
    6. 5.5 Substitution
    7. 5.6 Integrals Involving Exponential and Logarithmic Functions
    8. 5.7 Integrals Resulting in Inverse Trigonometric Functions
    9. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  7. 6 Applications of Integration
    1. Introduction
    2. 6.1 Areas between Curves
    3. 6.2 Determining Volumes by Slicing
    4. 6.3 Volumes of Revolution: Cylindrical Shells
    5. 6.4 Arc Length of a Curve and Surface Area
    6. 6.5 Physical Applications
    7. 6.6 Moments and Centers of Mass
    8. 6.7 Integrals, Exponential Functions, and Logarithms
    9. 6.8 Exponential Growth and Decay
    10. 6.9 Calculus of the Hyperbolic Functions
    11. Chapter Review
      1. Key Terms
      2. Key Equations
      3. Key Concepts
      4. Review Exercises
  8. A | Table of Integrals
  9. B | Table of Derivatives
  10. C | Review of Pre-Calculus
  11. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
  12. 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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