{"id":366,"date":"2024-10-01T18:45:49","date_gmt":"2024-10-01T18:45:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/appendix-a-table-of-integrals\/"},"modified":"2025-12-17T15:18:14","modified_gmt":"2025-12-17T15:18:14","slug":"appendix-a-table-of-integrals","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/appendix-a-table-of-integrals\/","title":{"raw":"Appendix A: Table of Integrals","rendered":"Appendix A: Table of Integrals"},"content":{"raw":"\n
\n

Basic Integrals<\/h2>\n

1. [latex]\\displaystyle\\int {u}^{n}du=\\frac{{u}^{n+1}}{n+1}+C,n\\ne \\text{\u2212}1[\/latex]<\/p>\n

2. [latex]\\displaystyle\\int \\frac{du}{u}=\\text{ln}|u|+C[\/latex]<\/p>\n

3. [latex]\\displaystyle\\int {e}^{u}du={e}^{u}+C[\/latex]<\/p>\n

4. [latex]\\displaystyle\\int {a}^{u}du=\\frac{{a}^{u}}{\\text{ln}a}+C[\/latex]<\/p>\n

5. [latex]\\displaystyle\\int \\sin udu=\\text{\u2212cos}u+C[\/latex]<\/p>\n

6. [latex]\\displaystyle\\int \\cos udu={\\sin}u+C[\/latex]<\/p>\n

7. [latex]\\displaystyle\\int { \\sec }^{2}udu= \\tan u+C[\/latex]<\/p>\n

8. [latex]\\displaystyle\\int { \\csc }^{2}udu=\\text{\u2212cot}u+C[\/latex]<\/p>\n

9. [latex]\\displaystyle\\int \\sec u \\tan udu= \\sec u+C[\/latex]<\/p>\n

10. [latex]\\displaystyle\\int \\csc u \\cot udu=\\text{\u2212csc}u+C[\/latex]<\/p>\n

11. [latex]\\displaystyle\\int \\tan udu=\\text{ln}| \\sec u|+C[\/latex]<\/p>\n

12. [latex]\\displaystyle\\int \\cot udu=\\text{ln}| \\sin u|+C[\/latex]<\/p>\n

13. [latex]\\displaystyle\\int \\sec udu=\\text{ln}| \\sec u+ \\tan u|+C[\/latex]<\/p>\n

14. [latex]\\displaystyle\\int \\csc udu=\\text{ln}| \\csc u- \\cot u|+C[\/latex]<\/p>\n

15. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

16. [latex]\\displaystyle\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}{ \\tan }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

17. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}{ \\sec }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n\n<\/div>\n

\n

Trigonometric Integrals<\/h2>\n

18. [latex]\\displaystyle\\int { \\sin }^{2}udu=\\frac{1}{2}u-\\frac{1}{4} \\sin 2u+C[\/latex]<\/p>\n

19. [latex]\\displaystyle\\int { \\cos }^{2}udu=\\frac{1}{2}u+\\frac{1}{4} \\sin 2u+C[\/latex]<\/p>\n

20. [latex]\\displaystyle\\int { \\tan }^{2}udu= \\tan u-u+C[\/latex]<\/p>\n

21. [latex]\\displaystyle\\int { \\cot }^{2}udu=\\text{\u2212} \\cot u-u+C[\/latex]<\/p>\n

22. [latex]\\displaystyle\\int { \\sin }^{3}udu=-\\frac{1}{3}(2+{ \\sin }^{2}u) \\cos u+C[\/latex]<\/p>\n

23. [latex]\\displaystyle\\int { \\cos }^{3}udu=\\frac{1}{3}(2+{ \\cos }^{2}u) \\sin u+C[\/latex]<\/p>\n

24. [latex]\\displaystyle\\int { \\tan }^{3}udu=\\frac{1}{2}{ \\tan }^{2}u+\\text{ln}| \\cos u|+C[\/latex]<\/p>\n

25. [latex]\\displaystyle\\int { \\cot }^{3}udu=-\\frac{1}{2}{ \\cot }^{2}u-\\text{ln}| \\sin u|+C[\/latex]<\/p>\n

26. [latex]\\displaystyle\\int { \\sec }^{3}udu=\\frac{1}{2} \\sec u \\tan u+\\frac{1}{2}\\text{ln}| \\sec u+ \\tan u|+C[\/latex]<\/p>\n

27. [latex]\\displaystyle\\int { \\csc }^{3}udu=-\\frac{1}{2} \\csc u \\cot u+\\frac{1}{2}\\text{ln}| \\csc u- \\cot u|+C[\/latex]<\/p>\n

28. [latex]\\displaystyle\\int { \\sin }^{n}udu=-\\frac{1}{n}{ \\sin }^{n-1}u \\cos u+\\frac{n-1}{n}\\displaystyle\\int { \\sin }^{n-2}udu[\/latex]<\/p>\n

29. [latex]\\displaystyle\\int { \\cos }^{n}udu=\\frac{1}{n}{ \\cos }^{n-1}u \\sin u+\\frac{n-1}{n}\\displaystyle\\int { \\cos }^{n-2}udu[\/latex]<\/p>\n

30. [latex]\\displaystyle\\int { \\tan }^{n}udu=\\frac{1}{n-1}{ \\tan }^{n-1}u-\\displaystyle\\int { \\tan }^{n-2}udu[\/latex]<\/p>\n

31. [latex]\\displaystyle\\int { \\cot }^{n}udu=\\frac{-1}{n-1}{ \\cot }^{n-1}u-\\displaystyle\\int { \\cot }^{n-2}udu[\/latex]<\/p>\n

32. [latex]\\displaystyle\\int { \\sec }^{n}udu=\\frac{1}{n-1} \\tan u{ \\sec }^{n-2}u+\\frac{n-2}{n-1}\\displaystyle\\int { \\sec }^{n-2}udu[\/latex]<\/p>\n

33. [latex]\\displaystyle\\int { \\csc }^{n}udu=\\frac{-1}{n-1} \\cot u{ \\csc }^{n-2}u+\\frac{n-2}{n-1}\\displaystyle\\int { \\csc }^{n-2}udu[\/latex]<\/p>\n

34. [latex]\\displaystyle\\int \\sin au \\sin budu=\\frac{ \\sin (a-b)u}{2(a-b)}-\\frac{ \\sin (a+b)u}{2(a+b)}+C[\/latex]<\/p>\n

35. [latex]\\displaystyle\\int \\cos au \\cos budu=\\frac{ \\sin (a-b)u}{2(a-b)}+\\frac{ \\sin (a+b)u}{2(a+b)}+C[\/latex]<\/p>\n

36. [latex]\\displaystyle\\int \\sin au \\cos budu=-\\frac{ \\cos (a-b)u}{2(a-b)}-\\frac{ \\cos (a+b)u}{2(a+b)}+C[\/latex]<\/p>\n

37. [latex]\\displaystyle\\int u \\sin udu= \\sin u-u \\cos u+C[\/latex]<\/p>\n

38. [latex]\\displaystyle\\int u \\cos udu= \\cos u+u \\sin u+C[\/latex]<\/p>\n

39. [latex]\\displaystyle\\int {u}^{n} \\sin udu=\\text{\u2212}{u}^{n} \\cos u+n\\displaystyle\\int {u}^{n-1} \\cos udu[\/latex]<\/p>\n

40. [latex]\\displaystyle\\int {u}^{n} \\cos udu={u}^{n} \\sin u-n\\displaystyle\\int {u}^{n-1} \\sin udu[\/latex]<\/p>\n

41. [latex]\\begin{array}{cc}\\hfill {\\displaystyle\\int { \\sin }^{n}u{ \\cos }^{m}udu}& =-\\frac{{ \\sin }^{n-1}u{ \\cos }^{m+1}u}{n+m}+\\frac{n-1}{n+m}{\\displaystyle\\int { \\sin }^{n-2}u{ \\cos }^{m}udu}\\hfill \\\\ & =\\frac{{ \\sin }^{n+1}u{ \\cos }^{m-1}u}{n+m}+\\frac{m-1}{n+m}{\\displaystyle\\int { \\sin }^{n}u{ \\cos }^{m-2}udu}\\hfill \\end{array}[\/latex]<\/p>\n\n<\/div>\n

\n

Exponential and Logarithmic Integrals<\/h2>\n

42. [latex]\\displaystyle\\int u{e}^{au}du=\\frac{1}{{a}^{2}}(au-1){e}^{au}+C[\/latex]<\/p>\n

43. [latex]\\displaystyle\\int {u}^{n}{e}^{au}du=\\frac{1}{a}{u}^{n}{e}^{au}-\\frac{n}{a}\\displaystyle\\int {u}^{n-1}{e}^{au}du[\/latex]<\/p>\n

44. [latex]\\displaystyle\\int {e}^{au} \\sin budu=\\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \\sin bu-b \\cos bu)+C[\/latex]<\/p>\n

45. [latex]\\displaystyle\\int {e}^{au} \\cos budu=\\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \\cos bu+b \\sin bu)+C[\/latex]<\/p>\n

46. [latex]\\displaystyle\\int \\text{ln}udu=u\\text{ln}u-u+C[\/latex]<\/p>\n

47. [latex]\\displaystyle\\int {u}^{n}\\text{ln}udu=\\frac{{u}^{n+1}}{{(n+1)}^{2}}\\left[(n+1)\\text{ln}u-1\\right]+C[\/latex]<\/p>\n

48. [latex]\\displaystyle\\int \\frac{1}{u\\text{ln}u}du=\\text{ln}|\\text{ln}u|+C[\/latex]<\/p>\n\n<\/div>\n

\n

Hyperbolic Integrals<\/h2>\n

49. [latex]\\displaystyle\\int \\text{sinh}udu=\\text{cosh}u+C[\/latex]<\/p>\n

50. [latex]\\displaystyle\\int \\text{cosh}udu=\\text{sinh}u+C[\/latex]<\/p>\n

51. [latex]\\displaystyle\\int \\text{tanh}udu=\\text{ln}\\text{cosh}u+C[\/latex]<\/p>\n

52. [latex]\\displaystyle\\int \\text{coth}udu=\\text{ln}|\\text{sinh}u|+C[\/latex]<\/p>\n

53. [latex]\\displaystyle\\int \\text{sech}udu={ \\tan }^{-1}|\\text{sinh}u|+C[\/latex]<\/p>\n

54. [latex]\\displaystyle\\int \\text{csch}udu=\\text{ln}|\\text{tanh}\\frac{1}{2}u|+C[\/latex]<\/p>\n

55. [latex]\\displaystyle\\int {\\text{sech}}^{2}udu=\\text{tanh}u+C[\/latex]<\/p>\n

56. [latex]\\displaystyle\\int {\\text{csch}}^{2}udu=\\text{\u2212}\\text{coth}u+C[\/latex]<\/p>\n

57. [latex]\\displaystyle\\int \\text{sech}u\\text{tanh}udu=\\text{\u2212}\\text{sech}u+C[\/latex]<\/p>\n

58. [latex]\\displaystyle\\int \\text{csch}u\\text{coth}udu=\\text{\u2212}\\text{csch}u+C[\/latex]<\/p>\n\n<\/div>\n

\n

Inverse Trigonometric Integrals<\/h2>\n

59. [latex]\\displaystyle\\int { \\sin }^{-1}udu=u{ \\sin }^{-1}u+\\sqrt{1-{u}^{2}}+C[\/latex]<\/p>\n

60. [latex]\\displaystyle\\int { \\cos }^{-1}udu=u{ \\cos }^{-1}u-\\sqrt{1-{u}^{2}}+C[\/latex]<\/p>\n

61. [latex]\\displaystyle\\int { \\tan }^{-1}udu=u{ \\tan }^{-1}u-\\frac{1}{2}\\text{ln}(1+{u}^{2})+C[\/latex]<\/p>\n

62. [latex]\\displaystyle\\int u{ \\sin }^{-1}udu=\\frac{2{u}^{2}-1}{4}{ \\sin }^{-1}u+\\frac{u\\sqrt{1-{u}^{2}}}{4}+C[\/latex]<\/p>\n

63. [latex]\\displaystyle\\int u{ \\cos }^{-1}udu=\\frac{2{u}^{2}-1}{4}{ \\cos }^{-1}u-\\frac{u\\sqrt{1-{u}^{2}}}{4}+C[\/latex]<\/p>\n

64. [latex]\\displaystyle\\int u{ \\tan }^{-1}udu=\\frac{{u}^{2}+1}{2}{ \\tan }^{-1}u-\\frac{u}{2}+C[\/latex]<\/p>\n

65. [latex]\\displaystyle\\int {u}^{n}{ \\sin }^{-1}udu=\\frac{1}{n+1}\\left[{u}^{n+1}{ \\sin }^{-1}u-\\displaystyle\\int \\frac{{u}^{n+1}du}{\\sqrt{1-{u}^{2}}}\\right],n\\ne \\text{\u2212}1[\/latex]<\/p>\n

66. [latex]\\displaystyle\\int {u}^{n}{ \\cos }^{-1}udu=\\frac{1}{n+1}\\left[{u}^{n+1}{ \\cos }^{-1}u+\\displaystyle\\int \\frac{{u}^{n+1}du}{\\sqrt{1-{u}^{2}}}\\right],n\\ne \\text{\u2212}1[\/latex]<\/p>\n

67. [latex]\\displaystyle\\int {u}^{n}{ \\tan }^{-1}udu=\\frac{1}{n+1}\\left[{u}^{n+1}{ \\tan }^{-1}u-\\displaystyle\\int \\frac{{u}^{n+1}du}{1+{u}^{2}}\\right],n\\ne \\text{\u2212}1[\/latex]<\/p>\n\n<\/div>\n

\n

Integrals Involving [latex]a[\/latex]2<\/sup> + [latex]u[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

68. [latex]\\displaystyle\\int \\sqrt{{a}^{2}+{u}^{2}}du=\\frac{u}{2}\\sqrt{{a}^{2}+{u}^{2}}+\\frac{{a}^{2}}{2}\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

69. [latex]\\displaystyle\\int {u}^{2}\\sqrt{{a}^{2}+{u}^{2}}du=\\frac{u}{8}({a}^{2}+2{u}^{2})\\sqrt{{a}^{2}+{u}^{2}}-\\frac{{a}^{4}}{8}\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

70. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}+{u}^{2}}}{u}du=\\sqrt{{a}^{2}+{u}^{2}}-a\\text{ln}|\\frac{a+\\sqrt{{a}^{2}+{u}^{2}}}{u}|+C[\/latex]<\/p>\n

71. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}+{u}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{a}^{2}+{u}^{2}}}{u}+\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

72. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{{a}^{2}+{u}^{2}}}=\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

73. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{{a}^{2}+{u}^{2}}}=\\frac{u}{2}(\\sqrt{{a}^{2}+{u}^{2}})-\\frac{{a}^{2}}{2}\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

74. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{a}^{2}+{u}^{2}}}=-\\frac{1}{a}\\text{ln}|\\frac{\\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C[\/latex]<\/p>\n

75. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}\\sqrt{{a}^{2}+{u}^{2}}}=-\\frac{\\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C[\/latex]<\/p>\n

76. [latex]\\displaystyle\\int \\frac{du}{{({a}^{2}+{u}^{2})}^{3\\text{\/}2}}=\\frac{u}{{a}^{2}\\sqrt{{a}^{2}+{u}^{2}}}+C[\/latex]<\/p>\n\n<\/div>\n

\n

Integrals Involving [latex]u[\/latex]2<\/sup> \u2212 [latex]a[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

77. [latex]\\displaystyle\\int \\sqrt{{u}^{2}-{a}^{2}}du=\\frac{u}{2}\\sqrt{{u}^{2}-{a}^{2}}-\\frac{{a}^{2}}{2}\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

78. [latex]\\displaystyle\\int {u}^{2}\\sqrt{{u}^{2}-{a}^{2}}du=\\frac{u}{8}(2{u}^{2}-{a}^{2})\\sqrt{{u}^{2}-{a}^{2}}-\\frac{{a}^{4}}{8}\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

79. [latex]\\displaystyle\\int \\frac{\\sqrt{{u}^{2}-{a}^{2}}}{u}du=\\sqrt{{u}^{2}-{a}^{2}}-a{ \\cos }^{-1}\\frac{a}{|u|}+C[\/latex]<\/p>\n

80. [latex]\\displaystyle\\int \\frac{\\sqrt{{u}^{2}-{a}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{u}^{2}-{a}^{2}}}{u}+\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

81. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{{u}^{2}-{a}^{2}}}=\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

82. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{u}{2}\\sqrt{{u}^{2}-{a}^{2}}+\\frac{{a}^{2}}{2}\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

83. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{\\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C[\/latex]<\/p>\n

84. [latex]\\displaystyle\\int \\frac{du}{{({u}^{2}-{a}^{2})}^{3\\text{\/}2}}=\\text{\u2212}\\frac{u}{{a}^{2}\\sqrt{{u}^{2}-{a}^{2}}}+C[\/latex]<\/p>\n\n<\/div>\n

\n

Integrals Involving [latex]a[\/latex]2<\/sup> \u2212 [latex]u[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

85. [latex]\\displaystyle\\int \\sqrt{{a}^{2}-{u}^{2}}du=\\frac{u}{2}\\sqrt{{a}^{2}-{u}^{2}}+\\frac{{a}^{2}}{2}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

86. [latex]\\displaystyle\\int {u}^{2}\\sqrt{{a}^{2}-{u}^{2}}du=\\frac{u}{8}(2{u}^{2}-{a}^{2})\\sqrt{{a}^{2}-{u}^{2}}+\\frac{{a}^{4}}{8}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

87. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{u}du=\\sqrt{{a}^{2}-{u}^{2}}-a\\text{ln}|\\frac{a+\\sqrt{{a}^{2}-{u}^{2}}}{u}|+C[\/latex]<\/p>\n

88. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\\frac{1}{u}\\sqrt{{a}^{2}-{u}^{2}}-{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

89. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{{a}^{2}-{u}^{2}}}=-\\frac{u}{u}\\sqrt{{a}^{2}-{u}^{2}}+\\frac{{a}^{2}}{2}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

90. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{a}^{2}-{u}^{2}}}=-\\frac{1}{a}\\text{ln}|\\frac{a+\\sqrt{{a}^{2}-{u}^{2}}}{u}|+C[\/latex]<\/p>\n

91. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}\\sqrt{{a}^{2}-{u}^{2}}}=-\\frac{1}{{a}^{2}u}\\sqrt{{a}^{2}-{u}^{2}}+C[\/latex]<\/p>\n

92. [latex]\\displaystyle\\int {({a}^{2}-{u}^{2})}^{3\\text{\/}2}du=-\\frac{u}{8}(2{u}^{2}-5{a}^{2})\\sqrt{{a}^{2}-{u}^{2}}+\\frac{3{a}^{4}}{8}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

93. [latex]\\displaystyle\\int \\frac{du}{{({a}^{2}-{u}^{2})}^{3\\text{\/}2}}=-\\frac{u}{{a}^{2}\\sqrt{{a}^{2}-{u}^{2}}}+C[\/latex]<\/p>\n\n<\/div>\n

\n

Integrals Involving 2au<\/em> \u2212 [latex]u[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

94. [latex]\\displaystyle\\int \\sqrt{2au-{u}^{2}}du=\\frac{u-a}{2}\\sqrt{2au-{u}^{2}}+\\frac{{a}^{2}}{2}{ \\cos }^{-1}(\\frac{a-u}{a})+C[\/latex]<\/p>\n

95. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{2au-{u}^{2}}}={ \\cos }^{-1}(\\frac{a-u}{a})+C[\/latex]<\/p>\n

96. [latex]\\displaystyle\\int u\\sqrt{2au-{u}^{2}}du=\\frac{2{u}^{2}-au-3{a}^{2}}{6}\\sqrt{2au-{u}^{2}}+\\frac{{a}^{3}}{2}{ \\cos }^{-1}(\\frac{a-u}{a})+C[\/latex]<\/p>\n

97. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{2au-{u}^{2}}}=-\\frac{\\sqrt{2au-{u}^{2}}}{au}+C[\/latex]<\/p>\n\n<\/div>\n

\n

Integrals Involving [latex]a[\/latex] + bu<\/em>, [latex]a[\/latex] \u2260 0<\/h2>\n

98. [latex]\\displaystyle\\int \\frac{udu}{a+bu}=\\frac{1}{{b}^{2}}(a+bu-a\\text{ln}|a+bu|)+C[\/latex]<\/p>\n

99. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{a+bu}=\\frac{1}{2{b}^{3}}\\left[{(a+bu)}^{2}-4a(a+bu)+2{a}^{2}\\text{ln}|a+bu|\\right]+C[\/latex]<\/p>\n

100. [latex]\\displaystyle\\int \\frac{du}{u(a+bu)}=\\frac{1}{a}\\text{ln}|\\frac{u}{a+bu}|+C[\/latex]<\/p>\n

101. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}(a+bu)}=-\\frac{1}{au}+\\frac{b}{{a}^{2}}\\text{ln}|\\frac{a+bu}{u}|+C[\/latex]<\/p>\n

102. [latex]\\displaystyle\\int \\frac{udu}{{(a+bu)}^{2}}=\\frac{a}{{b}^{2}(a+bu)}+\\frac{1}{{b}^{2}}\\text{ln}|a+bu|+C[\/latex]<\/p>\n

103. [latex]\\displaystyle\\int \\frac{udu}{u{(a+bu)}^{2}}=\\frac{1}{a(a+bu)}-\\frac{1}{{a}^{2}}\\text{ln}|\\frac{a+bu}{u}|+C[\/latex]<\/p>\n

104. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{{(a+bu)}^{2}}=\\frac{1}{{b}^{3}}(a+bu-\\frac{{a}^{2}}{a+bu}-2a\\text{ln}|a+bu|)+C[\/latex]<\/p>\n

105. [latex]\\displaystyle\\int u\\sqrt{a+bu}du=\\frac{2}{15{b}^{2}}(3bu-2a){(a+bu)}^{3\\text{\/}2}+C[\/latex]<\/p>\n

106. [latex]\\displaystyle\\int \\frac{udu}{\\sqrt{a+bu}}=\\frac{2}{3{b}^{2}}(bu-2a)\\sqrt{a+bu}+C[\/latex]<\/p>\n

107. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{a+bu}}=\\frac{2}{15{b}^{3}}(8{a}^{2}+3{b}^{2}{u}^{2}-4abu)\\sqrt{a+bu}+C[\/latex]<\/p>\n

108. [latex]\\begin{array}{ccc}\\hfill \\displaystyle\\int \\frac{du}{u\\sqrt{a+bu}}& =\\frac{1}{\\sqrt{a}}\\text{ln}|\\frac{\\sqrt{a+bu}-\\sqrt{a}}{\\sqrt{a+bu}+\\sqrt{a}}|+C,\\hfill & \\text{ if }a>0\\hfill \\\\ & =\\frac{2}{\\sqrt{\\text{\u2212}a}} \\tan -1\\sqrt{\\frac{a+bu}{\\text{\u2212}a}}+C,\\hfill & \\text{ if }a<0\\hfill \\end{array}[\/latex]<\/p>\n

109. [latex]\\displaystyle\\int \\frac{\\sqrt{a+bu}}{u}du=2\\sqrt{a+bu}+a\\displaystyle\\int \\frac{du}{u\\sqrt{a+bu}}[\/latex]<\/p>\n

110. [latex]\\displaystyle\\int \\frac{\\sqrt{a+bu}}{{u}^{2}}du=-\\frac{\\sqrt{a+bu}}{u}+\\frac{b}{2}\\displaystyle\\int \\frac{du}{u\\sqrt{a+bu}}[\/latex]<\/p>\n

111. [latex]\\displaystyle\\int {u}^{n}\\sqrt{a+bu}du=\\frac{2}{b(2n+3)}\\left[{u}^{n}{(a+bu)}^{3\\text{\/}2}-na\\displaystyle\\int {u}^{n-1}\\sqrt{a+bu}du\\right][\/latex]<\/p>\n

112. [latex]\\displaystyle\\int \\frac{{u}^{n}du}{\\sqrt{a+bu}}=\\frac{2{u}^{n}\\sqrt{a+bu}}{b(2n+1)}-\\frac{2na}{b(2n+1)}\\displaystyle\\int \\frac{{u}^{n-1}du}{\\sqrt{a+bu}}[\/latex]<\/p>\n

113. [latex]\\displaystyle\\int \\frac{du}{{u}^{n}\\sqrt{a+bu}}=-\\frac{\\sqrt{a+bu}}{a(n-1){u}^{n-1}}-\\frac{b(2n-3)}{2a(n-1)}\\displaystyle\\int \\frac{du}{{u}^{n-1}\\sqrt{a+bu}}[\/latex]<\/p>\n\n<\/div>\n","rendered":"

\n

Basic Integrals<\/h2>\n

1. [latex]\\displaystyle\\int {u}^{n}du=\\frac{{u}^{n+1}}{n+1}+C,n\\ne \\text{\u2212}1[\/latex]<\/p>\n

2. [latex]\\displaystyle\\int \\frac{du}{u}=\\text{ln}|u|+C[\/latex]<\/p>\n

3. [latex]\\displaystyle\\int {e}^{u}du={e}^{u}+C[\/latex]<\/p>\n

4. [latex]\\displaystyle\\int {a}^{u}du=\\frac{{a}^{u}}{\\text{ln}a}+C[\/latex]<\/p>\n

5. [latex]\\displaystyle\\int \\sin udu=\\text{\u2212cos}u+C[\/latex]<\/p>\n

6. [latex]\\displaystyle\\int \\cos udu={\\sin}u+C[\/latex]<\/p>\n

7. [latex]\\displaystyle\\int { \\sec }^{2}udu= \\tan u+C[\/latex]<\/p>\n

8. [latex]\\displaystyle\\int { \\csc }^{2}udu=\\text{\u2212cot}u+C[\/latex]<\/p>\n

9. [latex]\\displaystyle\\int \\sec u \\tan udu= \\sec u+C[\/latex]<\/p>\n

10. [latex]\\displaystyle\\int \\csc u \\cot udu=\\text{\u2212csc}u+C[\/latex]<\/p>\n

11. [latex]\\displaystyle\\int \\tan udu=\\text{ln}| \\sec u|+C[\/latex]<\/p>\n

12. [latex]\\displaystyle\\int \\cot udu=\\text{ln}| \\sin u|+C[\/latex]<\/p>\n

13. [latex]\\displaystyle\\int \\sec udu=\\text{ln}| \\sec u+ \\tan u|+C[\/latex]<\/p>\n

14. [latex]\\displaystyle\\int \\csc udu=\\text{ln}| \\csc u- \\cot u|+C[\/latex]<\/p>\n

15. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{{a}^{2}-{u}^{2}}}={ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

16. [latex]\\displaystyle\\int \\frac{du}{{a}^{2}+{u}^{2}}=\\frac{1}{a}{ \\tan }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

17. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{1}{a}{ \\sec }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n<\/div>\n

\n

Trigonometric Integrals<\/h2>\n

18. [latex]\\displaystyle\\int { \\sin }^{2}udu=\\frac{1}{2}u-\\frac{1}{4} \\sin 2u+C[\/latex]<\/p>\n

19. [latex]\\displaystyle\\int { \\cos }^{2}udu=\\frac{1}{2}u+\\frac{1}{4} \\sin 2u+C[\/latex]<\/p>\n

20. [latex]\\displaystyle\\int { \\tan }^{2}udu= \\tan u-u+C[\/latex]<\/p>\n

21. [latex]\\displaystyle\\int { \\cot }^{2}udu=\\text{\u2212} \\cot u-u+C[\/latex]<\/p>\n

22. [latex]\\displaystyle\\int { \\sin }^{3}udu=-\\frac{1}{3}(2+{ \\sin }^{2}u) \\cos u+C[\/latex]<\/p>\n

23. [latex]\\displaystyle\\int { \\cos }^{3}udu=\\frac{1}{3}(2+{ \\cos }^{2}u) \\sin u+C[\/latex]<\/p>\n

24. [latex]\\displaystyle\\int { \\tan }^{3}udu=\\frac{1}{2}{ \\tan }^{2}u+\\text{ln}| \\cos u|+C[\/latex]<\/p>\n

25. [latex]\\displaystyle\\int { \\cot }^{3}udu=-\\frac{1}{2}{ \\cot }^{2}u-\\text{ln}| \\sin u|+C[\/latex]<\/p>\n

26. [latex]\\displaystyle\\int { \\sec }^{3}udu=\\frac{1}{2} \\sec u \\tan u+\\frac{1}{2}\\text{ln}| \\sec u+ \\tan u|+C[\/latex]<\/p>\n

27. [latex]\\displaystyle\\int { \\csc }^{3}udu=-\\frac{1}{2} \\csc u \\cot u+\\frac{1}{2}\\text{ln}| \\csc u- \\cot u|+C[\/latex]<\/p>\n

28. [latex]\\displaystyle\\int { \\sin }^{n}udu=-\\frac{1}{n}{ \\sin }^{n-1}u \\cos u+\\frac{n-1}{n}\\displaystyle\\int { \\sin }^{n-2}udu[\/latex]<\/p>\n

29. [latex]\\displaystyle\\int { \\cos }^{n}udu=\\frac{1}{n}{ \\cos }^{n-1}u \\sin u+\\frac{n-1}{n}\\displaystyle\\int { \\cos }^{n-2}udu[\/latex]<\/p>\n

30. [latex]\\displaystyle\\int { \\tan }^{n}udu=\\frac{1}{n-1}{ \\tan }^{n-1}u-\\displaystyle\\int { \\tan }^{n-2}udu[\/latex]<\/p>\n

31. [latex]\\displaystyle\\int { \\cot }^{n}udu=\\frac{-1}{n-1}{ \\cot }^{n-1}u-\\displaystyle\\int { \\cot }^{n-2}udu[\/latex]<\/p>\n

32. [latex]\\displaystyle\\int { \\sec }^{n}udu=\\frac{1}{n-1} \\tan u{ \\sec }^{n-2}u+\\frac{n-2}{n-1}\\displaystyle\\int { \\sec }^{n-2}udu[\/latex]<\/p>\n

33. [latex]\\displaystyle\\int { \\csc }^{n}udu=\\frac{-1}{n-1} \\cot u{ \\csc }^{n-2}u+\\frac{n-2}{n-1}\\displaystyle\\int { \\csc }^{n-2}udu[\/latex]<\/p>\n

34. [latex]\\displaystyle\\int \\sin au \\sin budu=\\frac{ \\sin (a-b)u}{2(a-b)}-\\frac{ \\sin (a+b)u}{2(a+b)}+C[\/latex]<\/p>\n

35. [latex]\\displaystyle\\int \\cos au \\cos budu=\\frac{ \\sin (a-b)u}{2(a-b)}+\\frac{ \\sin (a+b)u}{2(a+b)}+C[\/latex]<\/p>\n

36. [latex]\\displaystyle\\int \\sin au \\cos budu=-\\frac{ \\cos (a-b)u}{2(a-b)}-\\frac{ \\cos (a+b)u}{2(a+b)}+C[\/latex]<\/p>\n

37. [latex]\\displaystyle\\int u \\sin udu= \\sin u-u \\cos u+C[\/latex]<\/p>\n

38. [latex]\\displaystyle\\int u \\cos udu= \\cos u+u \\sin u+C[\/latex]<\/p>\n

39. [latex]\\displaystyle\\int {u}^{n} \\sin udu=\\text{\u2212}{u}^{n} \\cos u+n\\displaystyle\\int {u}^{n-1} \\cos udu[\/latex]<\/p>\n

40. [latex]\\displaystyle\\int {u}^{n} \\cos udu={u}^{n} \\sin u-n\\displaystyle\\int {u}^{n-1} \\sin udu[\/latex]<\/p>\n

41. [latex]\\begin{array}{cc}\\hfill {\\displaystyle\\int { \\sin }^{n}u{ \\cos }^{m}udu}& =-\\frac{{ \\sin }^{n-1}u{ \\cos }^{m+1}u}{n+m}+\\frac{n-1}{n+m}{\\displaystyle\\int { \\sin }^{n-2}u{ \\cos }^{m}udu}\\hfill \\\\ & =\\frac{{ \\sin }^{n+1}u{ \\cos }^{m-1}u}{n+m}+\\frac{m-1}{n+m}{\\displaystyle\\int { \\sin }^{n}u{ \\cos }^{m-2}udu}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n

\n

Exponential and Logarithmic Integrals<\/h2>\n

42. [latex]\\displaystyle\\int u{e}^{au}du=\\frac{1}{{a}^{2}}(au-1){e}^{au}+C[\/latex]<\/p>\n

43. [latex]\\displaystyle\\int {u}^{n}{e}^{au}du=\\frac{1}{a}{u}^{n}{e}^{au}-\\frac{n}{a}\\displaystyle\\int {u}^{n-1}{e}^{au}du[\/latex]<\/p>\n

44. [latex]\\displaystyle\\int {e}^{au} \\sin budu=\\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \\sin bu-b \\cos bu)+C[\/latex]<\/p>\n

45. [latex]\\displaystyle\\int {e}^{au} \\cos budu=\\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \\cos bu+b \\sin bu)+C[\/latex]<\/p>\n

46. [latex]\\displaystyle\\int \\text{ln}udu=u\\text{ln}u-u+C[\/latex]<\/p>\n

47. [latex]\\displaystyle\\int {u}^{n}\\text{ln}udu=\\frac{{u}^{n+1}}{{(n+1)}^{2}}\\left[(n+1)\\text{ln}u-1\\right]+C[\/latex]<\/p>\n

48. [latex]\\displaystyle\\int \\frac{1}{u\\text{ln}u}du=\\text{ln}|\\text{ln}u|+C[\/latex]<\/p>\n<\/div>\n

\n

Hyperbolic Integrals<\/h2>\n

49. [latex]\\displaystyle\\int \\text{sinh}udu=\\text{cosh}u+C[\/latex]<\/p>\n

50. [latex]\\displaystyle\\int \\text{cosh}udu=\\text{sinh}u+C[\/latex]<\/p>\n

51. [latex]\\displaystyle\\int \\text{tanh}udu=\\text{ln}\\text{cosh}u+C[\/latex]<\/p>\n

52. [latex]\\displaystyle\\int \\text{coth}udu=\\text{ln}|\\text{sinh}u|+C[\/latex]<\/p>\n

53. [latex]\\displaystyle\\int \\text{sech}udu={ \\tan }^{-1}|\\text{sinh}u|+C[\/latex]<\/p>\n

54. [latex]\\displaystyle\\int \\text{csch}udu=\\text{ln}|\\text{tanh}\\frac{1}{2}u|+C[\/latex]<\/p>\n

55. [latex]\\displaystyle\\int {\\text{sech}}^{2}udu=\\text{tanh}u+C[\/latex]<\/p>\n

56. [latex]\\displaystyle\\int {\\text{csch}}^{2}udu=\\text{\u2212}\\text{coth}u+C[\/latex]<\/p>\n

57. [latex]\\displaystyle\\int \\text{sech}u\\text{tanh}udu=\\text{\u2212}\\text{sech}u+C[\/latex]<\/p>\n

58. [latex]\\displaystyle\\int \\text{csch}u\\text{coth}udu=\\text{\u2212}\\text{csch}u+C[\/latex]<\/p>\n<\/div>\n

\n

Inverse Trigonometric Integrals<\/h2>\n

59. [latex]\\displaystyle\\int { \\sin }^{-1}udu=u{ \\sin }^{-1}u+\\sqrt{1-{u}^{2}}+C[\/latex]<\/p>\n

60. [latex]\\displaystyle\\int { \\cos }^{-1}udu=u{ \\cos }^{-1}u-\\sqrt{1-{u}^{2}}+C[\/latex]<\/p>\n

61. [latex]\\displaystyle\\int { \\tan }^{-1}udu=u{ \\tan }^{-1}u-\\frac{1}{2}\\text{ln}(1+{u}^{2})+C[\/latex]<\/p>\n

62. [latex]\\displaystyle\\int u{ \\sin }^{-1}udu=\\frac{2{u}^{2}-1}{4}{ \\sin }^{-1}u+\\frac{u\\sqrt{1-{u}^{2}}}{4}+C[\/latex]<\/p>\n

63. [latex]\\displaystyle\\int u{ \\cos }^{-1}udu=\\frac{2{u}^{2}-1}{4}{ \\cos }^{-1}u-\\frac{u\\sqrt{1-{u}^{2}}}{4}+C[\/latex]<\/p>\n

64. [latex]\\displaystyle\\int u{ \\tan }^{-1}udu=\\frac{{u}^{2}+1}{2}{ \\tan }^{-1}u-\\frac{u}{2}+C[\/latex]<\/p>\n

65. [latex]\\displaystyle\\int {u}^{n}{ \\sin }^{-1}udu=\\frac{1}{n+1}\\left[{u}^{n+1}{ \\sin }^{-1}u-\\displaystyle\\int \\frac{{u}^{n+1}du}{\\sqrt{1-{u}^{2}}}\\right],n\\ne \\text{\u2212}1[\/latex]<\/p>\n

66. [latex]\\displaystyle\\int {u}^{n}{ \\cos }^{-1}udu=\\frac{1}{n+1}\\left[{u}^{n+1}{ \\cos }^{-1}u+\\displaystyle\\int \\frac{{u}^{n+1}du}{\\sqrt{1-{u}^{2}}}\\right],n\\ne \\text{\u2212}1[\/latex]<\/p>\n

67. [latex]\\displaystyle\\int {u}^{n}{ \\tan }^{-1}udu=\\frac{1}{n+1}\\left[{u}^{n+1}{ \\tan }^{-1}u-\\displaystyle\\int \\frac{{u}^{n+1}du}{1+{u}^{2}}\\right],n\\ne \\text{\u2212}1[\/latex]<\/p>\n<\/div>\n

\n

Integrals Involving [latex]a[\/latex]2<\/sup> + [latex]u[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

68. [latex]\\displaystyle\\int \\sqrt{{a}^{2}+{u}^{2}}du=\\frac{u}{2}\\sqrt{{a}^{2}+{u}^{2}}+\\frac{{a}^{2}}{2}\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

69. [latex]\\displaystyle\\int {u}^{2}\\sqrt{{a}^{2}+{u}^{2}}du=\\frac{u}{8}({a}^{2}+2{u}^{2})\\sqrt{{a}^{2}+{u}^{2}}-\\frac{{a}^{4}}{8}\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

70. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}+{u}^{2}}}{u}du=\\sqrt{{a}^{2}+{u}^{2}}-a\\text{ln}|\\frac{a+\\sqrt{{a}^{2}+{u}^{2}}}{u}|+C[\/latex]<\/p>\n

71. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}+{u}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{a}^{2}+{u}^{2}}}{u}+\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

72. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{{a}^{2}+{u}^{2}}}=\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

73. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{{a}^{2}+{u}^{2}}}=\\frac{u}{2}(\\sqrt{{a}^{2}+{u}^{2}})-\\frac{{a}^{2}}{2}\\text{ln}(u+\\sqrt{{a}^{2}+{u}^{2}})+C[\/latex]<\/p>\n

74. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{a}^{2}+{u}^{2}}}=-\\frac{1}{a}\\text{ln}|\\frac{\\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C[\/latex]<\/p>\n

75. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}\\sqrt{{a}^{2}+{u}^{2}}}=-\\frac{\\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C[\/latex]<\/p>\n

76. [latex]\\displaystyle\\int \\frac{du}{{({a}^{2}+{u}^{2})}^{3\\text{\/}2}}=\\frac{u}{{a}^{2}\\sqrt{{a}^{2}+{u}^{2}}}+C[\/latex]<\/p>\n<\/div>\n

\n

Integrals Involving [latex]u[\/latex]2<\/sup> \u2212 [latex]a[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

77. [latex]\\displaystyle\\int \\sqrt{{u}^{2}-{a}^{2}}du=\\frac{u}{2}\\sqrt{{u}^{2}-{a}^{2}}-\\frac{{a}^{2}}{2}\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

78. [latex]\\displaystyle\\int {u}^{2}\\sqrt{{u}^{2}-{a}^{2}}du=\\frac{u}{8}(2{u}^{2}-{a}^{2})\\sqrt{{u}^{2}-{a}^{2}}-\\frac{{a}^{4}}{8}\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

79. [latex]\\displaystyle\\int \\frac{\\sqrt{{u}^{2}-{a}^{2}}}{u}du=\\sqrt{{u}^{2}-{a}^{2}}-a{ \\cos }^{-1}\\frac{a}{|u|}+C[\/latex]<\/p>\n

80. [latex]\\displaystyle\\int \\frac{\\sqrt{{u}^{2}-{a}^{2}}}{{u}^{2}}du=-\\frac{\\sqrt{{u}^{2}-{a}^{2}}}{u}+\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

81. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{{u}^{2}-{a}^{2}}}=\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

82. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{u}{2}\\sqrt{{u}^{2}-{a}^{2}}+\\frac{{a}^{2}}{2}\\text{ln}|u+\\sqrt{{u}^{2}-{a}^{2}}|+C[\/latex]<\/p>\n

83. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}\\sqrt{{u}^{2}-{a}^{2}}}=\\frac{\\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C[\/latex]<\/p>\n

84. [latex]\\displaystyle\\int \\frac{du}{{({u}^{2}-{a}^{2})}^{3\\text{\/}2}}=\\text{\u2212}\\frac{u}{{a}^{2}\\sqrt{{u}^{2}-{a}^{2}}}+C[\/latex]<\/p>\n<\/div>\n

\n

Integrals Involving [latex]a[\/latex]2<\/sup> \u2212 [latex]u[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

85. [latex]\\displaystyle\\int \\sqrt{{a}^{2}-{u}^{2}}du=\\frac{u}{2}\\sqrt{{a}^{2}-{u}^{2}}+\\frac{{a}^{2}}{2}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

86. [latex]\\displaystyle\\int {u}^{2}\\sqrt{{a}^{2}-{u}^{2}}du=\\frac{u}{8}(2{u}^{2}-{a}^{2})\\sqrt{{a}^{2}-{u}^{2}}+\\frac{{a}^{4}}{8}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

87. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{u}du=\\sqrt{{a}^{2}-{u}^{2}}-a\\text{ln}|\\frac{a+\\sqrt{{a}^{2}-{u}^{2}}}{u}|+C[\/latex]<\/p>\n

88. [latex]\\displaystyle\\int \\frac{\\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\\frac{1}{u}\\sqrt{{a}^{2}-{u}^{2}}-{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

89. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{{a}^{2}-{u}^{2}}}=-\\frac{u}{u}\\sqrt{{a}^{2}-{u}^{2}}+\\frac{{a}^{2}}{2}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

90. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{{a}^{2}-{u}^{2}}}=-\\frac{1}{a}\\text{ln}|\\frac{a+\\sqrt{{a}^{2}-{u}^{2}}}{u}|+C[\/latex]<\/p>\n

91. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}\\sqrt{{a}^{2}-{u}^{2}}}=-\\frac{1}{{a}^{2}u}\\sqrt{{a}^{2}-{u}^{2}}+C[\/latex]<\/p>\n

92. [latex]\\displaystyle\\int {({a}^{2}-{u}^{2})}^{3\\text{\/}2}du=-\\frac{u}{8}(2{u}^{2}-5{a}^{2})\\sqrt{{a}^{2}-{u}^{2}}+\\frac{3{a}^{4}}{8}{ \\sin }^{-1}\\frac{u}{a}+C[\/latex]<\/p>\n

93. [latex]\\displaystyle\\int \\frac{du}{{({a}^{2}-{u}^{2})}^{3\\text{\/}2}}=-\\frac{u}{{a}^{2}\\sqrt{{a}^{2}-{u}^{2}}}+C[\/latex]<\/p>\n<\/div>\n

\n

Integrals Involving 2au<\/em> \u2212 [latex]u[\/latex]2<\/sup>, [latex]a[\/latex] > 0<\/h2>\n

94. [latex]\\displaystyle\\int \\sqrt{2au-{u}^{2}}du=\\frac{u-a}{2}\\sqrt{2au-{u}^{2}}+\\frac{{a}^{2}}{2}{ \\cos }^{-1}(\\frac{a-u}{a})+C[\/latex]<\/p>\n

95. [latex]\\displaystyle\\int \\frac{du}{\\sqrt{2au-{u}^{2}}}={ \\cos }^{-1}(\\frac{a-u}{a})+C[\/latex]<\/p>\n

96. [latex]\\displaystyle\\int u\\sqrt{2au-{u}^{2}}du=\\frac{2{u}^{2}-au-3{a}^{2}}{6}\\sqrt{2au-{u}^{2}}+\\frac{{a}^{3}}{2}{ \\cos }^{-1}(\\frac{a-u}{a})+C[\/latex]<\/p>\n

97. [latex]\\displaystyle\\int \\frac{du}{u\\sqrt{2au-{u}^{2}}}=-\\frac{\\sqrt{2au-{u}^{2}}}{au}+C[\/latex]<\/p>\n<\/div>\n

\n

Integrals Involving [latex]a[\/latex] + bu<\/em>, [latex]a[\/latex] \u2260 0<\/h2>\n

98. [latex]\\displaystyle\\int \\frac{udu}{a+bu}=\\frac{1}{{b}^{2}}(a+bu-a\\text{ln}|a+bu|)+C[\/latex]<\/p>\n

99. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{a+bu}=\\frac{1}{2{b}^{3}}\\left[{(a+bu)}^{2}-4a(a+bu)+2{a}^{2}\\text{ln}|a+bu|\\right]+C[\/latex]<\/p>\n

100. [latex]\\displaystyle\\int \\frac{du}{u(a+bu)}=\\frac{1}{a}\\text{ln}|\\frac{u}{a+bu}|+C[\/latex]<\/p>\n

101. [latex]\\displaystyle\\int \\frac{du}{{u}^{2}(a+bu)}=-\\frac{1}{au}+\\frac{b}{{a}^{2}}\\text{ln}|\\frac{a+bu}{u}|+C[\/latex]<\/p>\n

102. [latex]\\displaystyle\\int \\frac{udu}{{(a+bu)}^{2}}=\\frac{a}{{b}^{2}(a+bu)}+\\frac{1}{{b}^{2}}\\text{ln}|a+bu|+C[\/latex]<\/p>\n

103. [latex]\\displaystyle\\int \\frac{udu}{u{(a+bu)}^{2}}=\\frac{1}{a(a+bu)}-\\frac{1}{{a}^{2}}\\text{ln}|\\frac{a+bu}{u}|+C[\/latex]<\/p>\n

104. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{{(a+bu)}^{2}}=\\frac{1}{{b}^{3}}(a+bu-\\frac{{a}^{2}}{a+bu}-2a\\text{ln}|a+bu|)+C[\/latex]<\/p>\n

105. [latex]\\displaystyle\\int u\\sqrt{a+bu}du=\\frac{2}{15{b}^{2}}(3bu-2a){(a+bu)}^{3\\text{\/}2}+C[\/latex]<\/p>\n

106. [latex]\\displaystyle\\int \\frac{udu}{\\sqrt{a+bu}}=\\frac{2}{3{b}^{2}}(bu-2a)\\sqrt{a+bu}+C[\/latex]<\/p>\n

107. [latex]\\displaystyle\\int \\frac{{u}^{2}du}{\\sqrt{a+bu}}=\\frac{2}{15{b}^{3}}(8{a}^{2}+3{b}^{2}{u}^{2}-4abu)\\sqrt{a+bu}+C[\/latex]<\/p>\n

108. [latex]\\begin{array}{ccc}\\hfill \\displaystyle\\int \\frac{du}{u\\sqrt{a+bu}}& =\\frac{1}{\\sqrt{a}}\\text{ln}|\\frac{\\sqrt{a+bu}-\\sqrt{a}}{\\sqrt{a+bu}+\\sqrt{a}}|+C,\\hfill & \\text{ if }a>0\\hfill \\\\ & =\\frac{2}{\\sqrt{\\text{\u2212}a}} \\tan -1\\sqrt{\\frac{a+bu}{\\text{\u2212}a}}+C,\\hfill & \\text{ if }a<0\\hfill \\end{array}[\/latex]<\/p>\n

109. [latex]\\displaystyle\\int \\frac{\\sqrt{a+bu}}{u}du=2\\sqrt{a+bu}+a\\displaystyle\\int \\frac{du}{u\\sqrt{a+bu}}[\/latex]<\/p>\n

110. [latex]\\displaystyle\\int \\frac{\\sqrt{a+bu}}{{u}^{2}}du=-\\frac{\\sqrt{a+bu}}{u}+\\frac{b}{2}\\displaystyle\\int \\frac{du}{u\\sqrt{a+bu}}[\/latex]<\/p>\n

111. [latex]\\displaystyle\\int {u}^{n}\\sqrt{a+bu}du=\\frac{2}{b(2n+3)}\\left[{u}^{n}{(a+bu)}^{3\\text{\/}2}-na\\displaystyle\\int {u}^{n-1}\\sqrt{a+bu}du\\right][\/latex]<\/p>\n

112. [latex]\\displaystyle\\int \\frac{{u}^{n}du}{\\sqrt{a+bu}}=\\frac{2{u}^{n}\\sqrt{a+bu}}{b(2n+1)}-\\frac{2na}{b(2n+1)}\\displaystyle\\int \\frac{{u}^{n-1}du}{\\sqrt{a+bu}}[\/latex]<\/p>\n

113. [latex]\\displaystyle\\int \\frac{du}{{u}^{n}\\sqrt{a+bu}}=-\\frac{\\sqrt{a+bu}}{a(n-1){u}^{n-1}}-\\frac{b(2n-3)}{2a(n-1)}\\displaystyle\\int \\frac{du}{{u}^{n-1}\\sqrt{a+bu}}[\/latex]<\/p>\n<\/div>\n","protected":false},"author":6,"menu_order":2,"template":"","meta":{"_candela_citation":"{\"1\":{\"type\":\"cc\",\"description\":\"Calculus Volume 2\",\"author\":\"Gilbert Strang, Edwin (Jed) Herman\",\"organization\":\"OpenStax\",\"url\":\"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\",\"project\":\"\",\"license\":\"cc-by-nc-sa\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction\"}}","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":22,"module-header":"","content_attributions":{"1":{"type":"cc","description":"Calculus Volume 2","author":"Gilbert Strang, Edwin (Jed) Herman","organization":"OpenStax","url":"https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction","project":"","license":"cc-by-nc-sa","license_terms":"Access for free at https:\/\/openstax.org\/books\/calculus-volume-2\/pages\/1-introduction"}},"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/366"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":1,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/366\/revisions"}],"predecessor-version":[{"id":2507,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/366\/revisions\/2507"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/22"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/366\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=366"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=366"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=366"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=366"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}