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Study Guides > Calculus Volume 1

Table of Integrals

Basic Integrals

1. ∫undu=un+1n+1+C,n≠−1\int {u}^{n}du=\frac{{u}^{n+1}}{n+1}+C,n\ne \text{−}1

2. ∫duu=ln∣u∣+C\int \frac{du}{u}=\text{ln}|u|+C

3. ∫eudu=eu+C\int {e}^{u}du={e}^{u}+C

4. ∫audu=aulna+C\int {a}^{u}du=\frac{{a}^{u}}{\text{ln}a}+C

5. ∫sin⁡udu=−cosu+C\int \sin udu=\text{−cos}u+C

6. ∫cos⁡udu=sin⁡u+C\int \cos udu={\sin}u+C

7. ∫sec⁡2udu=tan⁡u+C\int { \sec }^{2}udu= \tan u+C

8. ∫csc⁡2udu=−cotu+C\int { \csc }^{2}udu=\text{−cot}u+C

9. ∫sec⁡utan⁡udu=sec⁡u+C\int \sec u \tan udu= \sec u+C

10. ∫csc⁡ucot⁡udu=−cscu+C\int \csc u \cot udu=\text{−csc}u+C

11. ∫tan⁡udu=ln∣sec⁡u∣+C\int \tan udu=\text{ln}| \sec u|+C

12. ∫cot⁡udu=ln∣sin⁡u∣+C\int \cot udu=\text{ln}| \sin u|+C

13. ∫sec⁡udu=ln∣sec⁡u+tan⁡u∣+C\int \sec udu=\text{ln}| \sec u+ \tan u|+C

14. ∫csc⁡udu=ln∣csc⁡u−cot⁡u∣+C\int \csc udu=\text{ln}| \csc u- \cot u|+C

15. ∫dua2−u2=sin⁡−1ua+C\int \frac{du}{\sqrt{{a}^{2}-{u}^{2}}}={ \sin }^{-1}\frac{u}{a}+C

16. ∫dua2+u2=1atan⁡−1ua+C\int \frac{du}{{a}^{2}+{u}^{2}}=\frac{1}{a}{ \tan }^{-1}\frac{u}{a}+C

17. ∫duuu2−a2=1asec⁡−1ua+C\int \frac{du}{u\sqrt{{u}^{2}-{a}^{2}}}=\frac{1}{a}{ \sec }^{-1}\frac{u}{a}+C

Trigonometric Integrals

18. ∫sin⁡2udu=12u−14sin⁡2u+C\int { \sin }^{2}udu=\frac{1}{2}u-\frac{1}{4} \sin 2u+C

19. ∫cos⁡2udu=12u+14sin⁡2u+C\int { \cos }^{2}udu=\frac{1}{2}u+\frac{1}{4} \sin 2u+C

20. ∫tan⁡2udu=tan⁡u−u+C\int { \tan }^{2}udu= \tan u-u+C

21. ∫cot⁡2udu=−cot⁡u−u+C\int { \cot }^{2}udu=\text{−} \cot u-u+C

22. ∫sin⁡3udu=−13(2+sin⁡2u)cos⁡u+C\int { \sin }^{3}udu=-\frac{1}{3}(2+{ \sin }^{2}u) \cos u+C

23. ∫cos⁡3udu=13(2+cos⁡2u)sin⁡u+C\int { \cos }^{3}udu=\frac{1}{3}(2+{ \cos }^{2}u) \sin u+C

24. ∫tan⁡3udu=12tan⁡2u+ln∣cos⁡u∣+C\int { \tan }^{3}udu=\frac{1}{2}{ \tan }^{2}u+\text{ln}| \cos u|+C

25. ∫cot⁡3udu=−12cot⁡2u−ln∣sin⁡u∣+C\int { \cot }^{3}udu=-\frac{1}{2}{ \cot }^{2}u-\text{ln}| \sin u|+C

26. ∫sec⁡3udu=12sec⁡utan⁡u+12ln∣sec⁡u+tan⁡u∣+C\int { \sec }^{3}udu=\frac{1}{2} \sec u \tan u+\frac{1}{2}\text{ln}| \sec u+ \tan u|+C

27. ∫csc⁡3udu=−12csc⁡ucot⁡u+12ln∣csc⁡u−cot⁡u∣+C\int { \csc }^{3}udu=-\frac{1}{2} \csc u \cot u+\frac{1}{2}\text{ln}| \csc u- \cot u|+C

28. ∫sin⁡nudu=−1nsin⁡n−1ucos⁡u+n−1n∫sin⁡n−2udu\int { \sin }^{n}udu=-\frac{1}{n}{ \sin }^{n-1}u \cos u+\frac{n-1}{n}\int { \sin }^{n-2}udu

29. ∫cos⁡nudu=1ncos⁡n−1usin⁡u+n−1n∫cos⁡n−2udu\int { \cos }^{n}udu=\frac{1}{n}{ \cos }^{n-1}u \sin u+\frac{n-1}{n}\int { \cos }^{n-2}udu

30. ∫tan⁡nudu=1n−1tan⁡n−1u−∫tan⁡n−2udu\int { \tan }^{n}udu=\frac{1}{n-1}{ \tan }^{n-1}u-\int { \tan }^{n-2}udu

31. ∫cot⁡nudu=−1n−1cot⁡n−1u−∫cot⁡n−2udu\int { \cot }^{n}udu=\frac{-1}{n-1}{ \cot }^{n-1}u-\int { \cot }^{n-2}udu

32. ∫sec⁡nudu=1n−1tan⁡usec⁡n−2u+n−2n−1∫sec⁡n−2udu\int { \sec }^{n}udu=\frac{1}{n-1} \tan u{ \sec }^{n-2}u+\frac{n-2}{n-1}\int { \sec }^{n-2}udu

33. ∫csc⁡nudu=−1n−1cot⁡ucsc⁡n−2u+n−2n−1∫csc⁡n−2udu\int { \csc }^{n}udu=\frac{-1}{n-1} \cot u{ \csc }^{n-2}u+\frac{n-2}{n-1}\int { \csc }^{n-2}udu

34. ∫sin⁡ausin⁡budu=sin⁡(a−b)u2(a−b)−sin⁡(a+b)u2(a+b)+C\int \sin au \sin budu=\frac{ \sin (a-b)u}{2(a-b)}-\frac{ \sin (a+b)u}{2(a+b)}+C

35. ∫cos⁡aucos⁡budu=sin⁡(a−b)u2(a−b)+sin⁡(a+b)u2(a+b)+C\int \cos au \cos budu=\frac{ \sin (a-b)u}{2(a-b)}+\frac{ \sin (a+b)u}{2(a+b)}+C

36. ∫sin⁡aucos⁡budu=−cos⁡(a−b)u2(a−b)−cos⁡(a+b)u2(a+b)+C\int \sin au \cos budu=-\frac{ \cos (a-b)u}{2(a-b)}-\frac{ \cos (a+b)u}{2(a+b)}+C

37. ∫usin⁡udu=sin⁡u−ucos⁡u+C\int u \sin udu= \sin u-u \cos u+C

38. ∫ucos⁡udu=cos⁡u+usin⁡u+C\int u \cos udu= \cos u+u \sin u+C

39. ∫unsin⁡udu=−uncos⁡u+n∫un−1cos⁡udu\int {u}^{n} \sin udu=\text{−}{u}^{n} \cos u+n\int {u}^{n-1} \cos udu

40. ∫uncos⁡udu=unsin⁡u−n∫un−1sin⁡udu\int {u}^{n} \cos udu={u}^{n} \sin u-n\int {u}^{n-1} \sin udu

41. ∫sin⁡nucos⁡mudu=−sin⁡n−1ucos⁡m+1un+m+n−1n+m∫sin⁡n−2ucos⁡mudu=sin⁡n+1ucos⁡m−1un+m+m−1n+m∫sin⁡nucos⁡m−2udu\begin{array}{cc}\hfill \int { \sin }^{n}u{ \cos }^{m}udu& =-\frac{{ \sin }^{n-1}u{ \cos }^{m+1}u}{n+m}+\frac{n-1}{n+m}\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}\int { \sin }^{n}u{ \cos }^{m-2}udu\hfill \end{array}

Exponential and Logarithmic Integrals

42. ∫ueaudu=1a2(au−1)eau+C\int u{e}^{au}du=\frac{1}{{a}^{2}}(au-1){e}^{au}+C

43. ∫uneaudu=1auneau−na∫un−1eaudu\int {u}^{n}{e}^{au}du=\frac{1}{a}{u}^{n}{e}^{au}-\frac{n}{a}\int {u}^{n-1}{e}^{au}du

44. ∫eausin⁡budu=eaua2+b2(asin⁡bu−bcos⁡bu)+C\int {e}^{au} \sin budu=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \sin bu-b \cos bu)+C

45. ∫eaucos⁡budu=eaua2+b2(acos⁡bu+bsin⁡bu)+C\int {e}^{au} \cos budu=\frac{{e}^{au}}{{a}^{2}+{b}^{2}}(a \cos bu+b \sin bu)+C

46. ∫lnudu=ulnu−u+C\int \text{ln}udu=u\text{ln}u-u+C

47. ∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C\int {u}^{n}\text{ln}udu=\frac{{u}^{n+1}}{{(n+1)}^{2}}\left[(n+1)\text{ln}u-1\right]+C

48. ∫1ulnudu=ln∣lnu∣+C\int \frac{1}{u\text{ln}u}du=\text{ln}|\text{ln}u|+C

Hyperbolic Integrals

49. ∫sinhudu=coshu+C\int \text{sinh}udu=\text{cosh}u+C

50. ∫coshudu=sinhu+C\int \text{cosh}udu=\text{sinh}u+C

51. ∫tanhudu=lncoshu+C\int \text{tanh}udu=\text{ln}\text{cosh}u+C

52. ∫cothudu=ln∣sinhu∣+C\int \text{coth}udu=\text{ln}|\text{sinh}u|+C

53. ∫sechudu=tan⁡−1∣sinhu∣+C\int \text{sech}udu={ \tan }^{-1}|\text{sinh}u|+C

54. ∫cschudu=ln∣tanh12u∣+C\int \text{csch}udu=\text{ln}|\text{tanh}\frac{1}{2}u|+C

55. ∫sech2udu=tanhu+C\int {\text{sech}}^{2}udu=\text{tanh}u+C

56. ∫csch2udu=−cothu+C\int {\text{csch}}^{2}udu=\text{−}\text{coth}u+C

57. ∫sechutanhudu=−sechu+C\int \text{sech}u\text{tanh}udu=\text{−}\text{sech}u+C

58. ∫cschucothudu=−cschu+C\int \text{csch}u\text{coth}udu=\text{−}\text{csch}u+C

Inverse Trigonometric Integrals

59. ∫sin⁡−1udu=usin⁡−1u+1−u2+C\int { \sin }^{-1}udu=u{ \sin }^{-1}u+\sqrt{1-{u}^{2}}+C

60. ∫cos⁡−1udu=ucos⁡−1u−1−u2+C\int { \cos }^{-1}udu=u{ \cos }^{-1}u-\sqrt{1-{u}^{2}}+C

61. ∫tan⁡−1udu=utan⁡−1u−12ln(1+u2)+C\int { \tan }^{-1}udu=u{ \tan }^{-1}u-\frac{1}{2}\text{ln}(1+{u}^{2})+C

62. ∫usin⁡−1udu=2u2−14sin⁡−1u+u1−u24+C\int u{ \sin }^{-1}udu=\frac{2{u}^{2}-1}{4}{ \sin }^{-1}u+\frac{u\sqrt{1-{u}^{2}}}{4}+C

63. ∫ucos⁡−1udu=2u2−14cos⁡−1u−u1−u24+C\int u{ \cos }^{-1}udu=\frac{2{u}^{2}-1}{4}{ \cos }^{-1}u-\frac{u\sqrt{1-{u}^{2}}}{4}+C

64. ∫utan⁡−1udu=u2+12tan⁡−1u−u2+C\int u{ \tan }^{-1}udu=\frac{{u}^{2}+1}{2}{ \tan }^{-1}u-\frac{u}{2}+C

65. ∫unsin⁡−1udu=1n+1[un+1sin⁡−1u−∫un+1du1−u2],n≠−1\int {u}^{n}{ \sin }^{-1}udu=\frac{1}{n+1}\left[{u}^{n+1}{ \sin }^{-1}u-\int \frac{{u}^{n+1}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1

66. ∫uncos⁡−1udu=1n+1[un+1cos⁡−1u+∫un+1du1−u2],n≠−1\int {u}^{n}{ \cos }^{-1}udu=\frac{1}{n+1}\left[{u}^{n+1}{ \cos }^{-1}u+\int \frac{{u}^{n+1}du}{\sqrt{1-{u}^{2}}}\right],n\ne \text{−}1

67. ∫untan⁡−1udu=1n+1[un+1tan⁡−1u−∫un+1du1+u2],n≠−1\int {u}^{n}{ \tan }^{-1}udu=\frac{1}{n+1}\left[{u}^{n+1}{ \tan }^{-1}u-\int \frac{{u}^{n+1}du}{1+{u}^{2}}\right],n\ne \text{−}1

Integrals Involving aa2 + uu2, aa > 0

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

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

70. ∫a2+u2udu=a2+u2−aln∣a+a2+u2u∣+C\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

71. ∫a2+u2u2du=−a2+u2u+ln(u+a2+u2)+C\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

72. ∫dua2+u2=ln(u+a2+u2)+C\int \frac{du}{\sqrt{{a}^{2}+{u}^{2}}}=\text{ln}(u+\sqrt{{a}^{2}+{u}^{2}})+C

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

74. ∫duua2+u2=−1aln∣a2+u2+au∣+C\int \frac{du}{u\sqrt{{a}^{2}+{u}^{2}}}=-\frac{1}{a}\text{ln}|\frac{\sqrt{{a}^{2}+{u}^{2}}+a}{u}|+C

75. ∫duu2a2+u2=−a2+u2a2u+C\int \frac{du}{{u}^{2}\sqrt{{a}^{2}+{u}^{2}}}=-\frac{\sqrt{{a}^{2}+{u}^{2}}}{{a}^{2}u}+C

76. ∫du(a2+u2)3/2=ua2a2+u2+C\int \frac{du}{{({a}^{2}+{u}^{2})}^{3\text{/}2}}=\frac{u}{{a}^{2}\sqrt{{a}^{2}+{u}^{2}}}+C

Integrals Involving uu2 − aa2, aa > 0

77. ∫u2−a2du=u2u2−a2−a22ln∣u+u2−a2∣+C\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

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

79. ∫u2−a2udu=u2−a2−acos⁡−1a∣u∣+C\int \frac{\sqrt{{u}^{2}-{a}^{2}}}{u}du=\sqrt{{u}^{2}-{a}^{2}}-a{ \cos }^{-1}\frac{a}{|u|}+C

80. ∫u2−a2u2du=−u2−a2u+ln∣u+u2−a2∣+C\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

81. ∫duu2−a2=ln∣u+u2−a2∣+C\int \frac{du}{\sqrt{{u}^{2}-{a}^{2}}}=\text{ln}|u+\sqrt{{u}^{2}-{a}^{2}}|+C

82. ∫u2duu2−a2=u2u2−a2+a22ln∣u+u2−a2∣+C\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

83. ∫duu2u2−a2=u2−a2a2u+C\int \frac{du}{{u}^{2}\sqrt{{u}^{2}-{a}^{2}}}=\frac{\sqrt{{u}^{2}-{a}^{2}}}{{a}^{2}u}+C

84. ∫du(u2−a2)3/2=−ua2u2−a2+C\int \frac{du}{{({u}^{2}-{a}^{2})}^{3\text{/}2}}=\text{−}\frac{u}{{a}^{2}\sqrt{{u}^{2}-{a}^{2}}}+C

Integrals Involving aa2 − uu2, aa > 0

85. ∫a2−u2du=u2a2−u2+a22sin⁡−1ua+C\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

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

87. ∫a2−u2udu=a2−u2−aln∣a+a2−u2u∣+C\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

88. ∫a2−u2u2du=−1ua2−u2−sin⁡−1ua+C\int \frac{\sqrt{{a}^{2}-{u}^{2}}}{{u}^{2}}du=-\frac{1}{u}\sqrt{{a}^{2}-{u}^{2}}-{ \sin }^{-1}\frac{u}{a}+C

89. ∫u2dua2−u2=−uua2−u2+a22sin⁡−1ua+C\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

90. ∫duua2−u2=−1aln∣a+a2−u2u∣+C\int \frac{du}{u\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{a}\text{ln}|\frac{a+\sqrt{{a}^{2}-{u}^{2}}}{u}|+C

91. ∫duu2a2−u2=−1a2ua2−u2+C\int \frac{du}{{u}^{2}\sqrt{{a}^{2}-{u}^{2}}}=-\frac{1}{{a}^{2}u}\sqrt{{a}^{2}-{u}^{2}}+C

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

93. ∫du(a2−u2)3/2=−ua2a2−u2+C\int \frac{du}{{({a}^{2}-{u}^{2})}^{3\text{/}2}}=-\frac{u}{{a}^{2}\sqrt{{a}^{2}-{u}^{2}}}+C

Integrals Involving 2au − uu2, aa > 0

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

95. ∫du2au−u2=cos⁡−1(a−ua)+C\int \frac{du}{\sqrt{2au-{u}^{2}}}={ \cos }^{-1}(\frac{a-u}{a})+C

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

97. ∫duu2au−u2=−2au−u2au+C\int \frac{du}{u\sqrt{2au-{u}^{2}}}=-\frac{\sqrt{2au-{u}^{2}}}{au}+C

Integrals Involving aa + bu, aa ≠ 0

98. ∫udua+bu=1b2(a+bu−aln∣a+bu∣)+C\int \frac{udu}{a+bu}=\frac{1}{{b}^{2}}(a+bu-a\text{ln}|a+bu|)+C

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

100. ∫duu(a+bu)=1aln∣ua+bu∣+C\int \frac{du}{u(a+bu)}=\frac{1}{a}\text{ln}|\frac{u}{a+bu}|+C

101. ∫duu2(a+bu)=−1au+ba2ln∣a+buu∣+C\int \frac{du}{{u}^{2}(a+bu)}=-\frac{1}{au}+\frac{b}{{a}^{2}}\text{ln}|\frac{a+bu}{u}|+C

102. ∫udu(a+bu)2=ab2(a+bu)+1b2ln∣a+bu∣+C\int \frac{udu}{{(a+bu)}^{2}}=\frac{a}{{b}^{2}(a+bu)}+\frac{1}{{b}^{2}}\text{ln}|a+bu|+C

103. ∫uduu(a+bu)2=1a(a+bu)−1a2ln∣a+buu∣+C\int \frac{udu}{u{(a+bu)}^{2}}=\frac{1}{a(a+bu)}-\frac{1}{{a}^{2}}\text{ln}|\frac{a+bu}{u}|+C

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

105. ∫ua+budu=215b2(3bu−2a)(a+bu)3/2+C\int u\sqrt{a+bu}du=\frac{2}{15{b}^{2}}(3bu-2a){(a+bu)}^{3\text{/}2}+C

106. ∫udua+bu=23b2(bu−2a)a+bu+C\int \frac{udu}{\sqrt{a+bu}}=\frac{2}{3{b}^{2}}(bu-2a)\sqrt{a+bu}+C

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

108. ∫duua+bu=1aln∣a+bu−aa+bu+a∣+C, if a>0=2−atan⁡−1a+bu−a+C, if a<0\begin{array}{ccc}\hfill \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{−}a}} \tan -1\sqrt{\frac{a+bu}{\text{−}a}}+C,\hfill & \text{ if }a<0\hfill \end{array}

109. ∫a+buudu=2a+bu+a∫duua+bu\int \frac{\sqrt{a+bu}}{u}du=2\sqrt{a+bu}+a\int \frac{du}{u\sqrt{a+bu}}

110. ∫a+buu2du=−a+buu+b2∫duua+bu\int \frac{\sqrt{a+bu}}{{u}^{2}}du=-\frac{\sqrt{a+bu}}{u}+\frac{b}{2}\int \frac{du}{u\sqrt{a+bu}}

111. ∫una+budu=2b(2n+3)[un(a+bu)3/2−na∫un−1a+budu]\int {u}^{n}\sqrt{a+bu}du=\frac{2}{b(2n+3)}\left[{u}^{n}{(a+bu)}^{3\text{/}2}-na\int {u}^{n-1}\sqrt{a+bu}du\right]

112. ∫undua+bu=2una+bub(2n+1)−2nab(2n+1)∫un−1dua+bu\int \frac{{u}^{n}du}{\sqrt{a+bu}}=\frac{2{u}^{n}\sqrt{a+bu}}{b(2n+1)}-\frac{2na}{b(2n+1)}\int \frac{{u}^{n-1}du}{\sqrt{a+bu}}

113. ∫duuna+bu=−a+bua(n−1)un−1−b(2n−3)2a(n−1)∫duun−1a+bu\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)}\int \frac{du}{{u}^{n-1}\sqrt{a+bu}}

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