18.2: Table of Integrals
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- Dec 29, 2024
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- Gilbert Strang & Edwin “Jed” Herman
- OpenStax
( \newcommand{\kernel}{\mathrm{null}\,}\)
Basic Integrals
1. ∫undu=un+1n+1+C,n≠−1
2. ∫duu=ln|u|+C
3. ∫eudu=eu+C
4. ∫audu=aulna+C
5. ∫sinudu=−cosu+C
6. ∫cosudu=sinu+C
7. ∫sec2udu=tanu+C
8. ∫csc2udu=−cotu+C
9. ∫secutanudu=secu+C
10. ∫cscucotudu=−cscu+C
11. ∫tanudu=ln|secu|+C
12. ∫cotudu=ln|sinu|+C
13. ∫secudu=ln|secu+tanu|+C
14. ∫cscudu=ln|cscu−cotu|+C
15. ∫du√a2−u2=sin−1(ua)+C
16. ∫dua2+u2=1atan−1(ua)+C
17. ∫duu√u2−a2=1asec−1|u|a+C
Trigonometric Integrals
18. ∫sin2udu=12u−14sin2u+C
19. ∫cos2udu=12u+14sin2u+C
20. ∫tan2udu=tanu−u+C
21. ∫cot2udu=−cotu−u+C
22. ∫sin3udu=−13(2+sin2u)cosu+C
23. ∫cos3udu=13(2+cos2u)sinu+C
24. ∫tan3udu=12tan2u+ln|cosu|+C
25. ∫cot3udu=−12cot2u−ln|sinu|+C
26. ∫sec3udu=12secutanu+12ln|secu+tanu|+C
27. ∫csc3udu=−12cscucotu+12ln|cscu−cotu|+C
28. ∫sinnudu=−1nsinn−1ucosu+n−1n∫sinn−2udu
29. ∫cosnudu=1ncosn−1usinu+n−1n∫cosn−2udu
30. ∫tannudu=1n−1tann−1u−∫tann−2udu
31. ∫cotnudu=−1n−1cotn−1u−∫cotn−2udu
32. ∫secnudu=1n−1tanusecn−2u+n−2n−1∫secn−2udu
33. ∫cscnudu=−1n−1cotucscn−2u+n−2n−1∫cscn−2udu
34. ∫sinausinbudu=sin(a−b)u2(a−b)−sin(a+b)u2(a+b)+C
35. ∫cosaucosbudu=sin(a−b)u2(a−b)+sin(a+b)u2(a+b)+C
36. ∫sinaucosbudu=−cos(a−b)u2(a−b)−cos(a+b)u2(a+b)+C
37. ∫usinudu=sinu−ucosu+C
38. ∫ucosudu=cosu+usinu+C
39. ∫unsinudu=−uncosu+n∫un−1cosudu
40. ∫uncosudu=unsinu−n∫un−1sinudu
41. ∫sinnucosmudu=−sinn−1ucosm+1un+m+n−1n+m∫sinn−2ucosmudu=sinn+1ucosm−1un+m+m−1n+m∫sinnucosm−2udu
Exponential and Logarithmic Integrals
42. ∫ueaudu=1a2(au−1)eau+C
43. ∫uneaudu=1auneau−na∫un−1eaudu
44. ∫eausinbudu=eaua2+b2(asinbu−bcosbu)+C
45. ∫eaucosbudu=eaua2+b2(acosbu+bsinbu)+C
46. ∫lnudu=ulnu−u+C
47. ∫unlnudu=un+1(n+1)2[(n+1)lnu−1]+C
48. ∫1ulnudu=ln|lnu|+C
Hyperbolic Integrals
49. ∫sinhudu=coshu+C
50. ∫coshudu=sinhu+C
51. ∫tanhudu=lncoshu+C
52. ∫cothudu=ln|sinhu|+C
53. ∫sechudu=tan−1|sinhu|+C
54. ∫cschudu=ln∣tanh12u∣+C
55. ∫sech2udu=tanhu+C
56. ∫csch2udu=−cothu+C
57. ∫sechutanhudu=−sechu+C
58. ∫cschucothudu=−cschu+C
Inverse Trigonometric Integrals
59. ∫sin−1udu=usin−1u+√1−u2+C
60. ∫cos−1udu=ucos−1u−√1−u2+C
61. ∫tan−1udu=utan−1u−12ln(1+u2)+C
62. ∫usin−1udu=2u2−14sin−1u+u√1−u24+C
63. ∫ucos−1udu=2u2−14cos−1u−u√1−u24+C
64. ∫utan−1udu=u2+12tan−1u−u2+C
65. ∫unsin−1udu=1n+1[un+1sin−1u−∫un+1du√1−u2],n≠−1
66. ∫uncos−1udu=1n+1[un+1cos−1u+∫un+1du√1−u2],n≠−1
67. ∫untan−1udu=1n+1[un+1tan−1u−∫un+1du1+u2],n≠−1
Integrals Involving a2 + u2, a > 0
68. ∫√a2+u2du=u2√a2+u2+a22ln(u+√a2+u2)+C
69. ∫u2√a2+u2du=u8(a2+2u2)√a2+u2−a48ln(u+√a2+u2)+C
70. ∫√a2+u2udu=√a2+u2−aln|a+√a2+u2u|+C
71. ∫√a2+u2u2du=−√a2+u2u+ln(u+√a2+u2)+C
72. ∫du√a2+u2=ln(u+√a2+u2)+C
73. ∫u2√a2+u2du=u2(√a2+u2)−a22ln(u+√a2+u2)+C
74. ∫duu√a2+u2=−1aln|√a2+u2+au|+C
75. ∫duu2√a2+u2=−√a2+u2a2u+C
76. ∫du(a2+u2)3/2=ua2√a2+u2+C
Integrals Involving u2 − a2, a > 0
77. ∫√u2−a2du=u2√u2−a2−a22ln|u+√u2−a2|+C
78. ∫u2√u2−a2du=u8(2u2−a2)√u2−a2−a48ln|u+√u2−a2|+C
79. ∫√u2−a2udu=√u2−a2−acos−1a|u|+C
80. ∫√u2−a2u2du=−√u2−a2u+ln|u+√u2−a2|+C
81. ∫du√u2−a2=ln|u+√u2−a2|+C
82. ∫u2√u2−a2du=u2√u2−a2+a22ln|u+√u2−a2|+C
83. ∫duu2√u2−a2=√u2−a2a2u+C
84. ∫du(u2−a2)3/2=−ua2√u2−a2+C
Integrals Involving a2 − u2, a > 0
85. ∫√a2−u2du=u2√a2−u2+a22sin−1ua+C
86. ∫u2√a2−u2du=u8(2u2−a2)√a2−u2+a48sin−1ua+C
87. ∫√a2−u2udu=√a2−u2−aln|a+√a2−u2u|+C
88. ∫√a2−u2u2du=−1u√a2−u2−sin−1ua+C
89. ∫u2√a2−u2du=12(−u√a2−u2+a2sin−1ua)+C
90. ∫duu√a2−u2=−1aln|a+√a2−u2u|+C
91. ∫duu2√a2−u2=−1a2u√a2−u2+C
92. ∫(a2−u2)3/2du=−u8(2u2−5a2)√a2−u2+3a48sin−1ua+C
93. ∫du(a2−u2)3/2=−ua2√a2−u2+C
Integrals Involving 2au − u2, a > 0
94. ∫√2au−u2du=u−a2√2au−u2+a22cos−1(a−ua)+C
95. ∫du√2au−u2=cos−1(a−ua)+C
96. ∫u√2au−u2du=2u2−au−3a26√2au−u2+a32cos−1(a−ua)+C
97. ∫duu√2au−u2=−√2au−u2au+C
Integrals Involving a + bu, a ≠ 0
98. ∫ua+budu=1b2(a+bu−aln|a+bu|)+C
99. ∫u2a+budu=12b3[(a+bu)2−4a(a+bu)+2a2ln|a+bu|]+C
100. ∫duu(a+bu)=1aln|ua+bu|+C
101. \quad \displaystyle ∫\frac{du}{u^2(a+bu)}=−\frac{1}{au}+\frac{b}{a^2}\ln \left|\frac{a+bu}{u}\right|+C
102. \quad \displaystyle ∫\frac{u}{(a+bu)^2}\,du=\frac{a}{b^2(a+bu)}+\frac{1}{b^2}\ln |a+bu|+C
103. \quad \displaystyle ∫\frac{u}{u(a+bu)^2}\,du=\frac{1}{a(a+bu)}−\frac{1}{a^2}\ln \left|\frac{a+bu}{u}\right|+C
104. \quad \displaystyle ∫\frac{u^2}{(a+bu)^2}\,du=\frac{1}{b^3}\left(a+bu−\frac{a^2}{a+bu}−2a\ln |a+bu|\right)+C
105. \quad \displaystyle ∫u\sqrt{a+bu}\,du=\frac{2}{15b^2}(3bu−2a)(a+bu)^{3/2}+C
106. \quad \displaystyle ∫\frac{u}{\sqrt{a+bu}}\,du=\frac{2}{3b^2}(bu−2a)\sqrt{a+bu}+C
107. \quad \displaystyle ∫\frac{u^2}{\sqrt{a+bu}}\,du=\frac{2}{15b^3}(8a^2+3b^2u^2−4abu)\sqrt{a+bu}+C
108. \quad \displaystyle ∫\frac{du}{u\sqrt{a+bu}}=\begin{cases} \frac{1}{\sqrt{a}}\ln \left|\frac{\sqrt{a+bu}−\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}\right|+C,\quad \text{if}\,a>0\\[4pt] \frac{\sqrt{2}}{\sqrt{−a}}\tan^{-1}\sqrt{\frac{a+bu}{−a}}+C,\quad \text{if}\,a<0 \end{cases}
109. \quad \displaystyle ∫\frac{\sqrt{a+bu}}{u}\,du=2\sqrt{a+bu}+a∫\frac{du}{u\sqrt{a+bu}}
110. \quad \displaystyle ∫\frac{\sqrt{a+bu}}{u^2}\,du=−\frac{\sqrt{a+bu}}{u}+\frac{b}{2}∫\frac{du}{u\sqrt{a+bu}}
111. \quad \displaystyle ∫u^n\sqrt{a+bu}\,du=\frac{2}{b(2n+3)}\left[u^n(a+bu)^{3/2}−na∫u^{n−1}\sqrt{a+bu}\,du\right]
112. \quad \displaystyle ∫\frac{u^n}{\sqrt{a+bu}}\,du=\frac{2u^n\sqrt{a+bu}}{b(2n+1)}−\frac{2na}{b(2n+1)}∫\frac{u^{n−1}}{\sqrt{a+bu}}\,du
113. \quad \displaystyle ∫\frac{du}{u^n\sqrt{a+bu}}=−\frac{\sqrt{a+bu}}{a(n−1)u^{n−1}}−\frac{b(2n−3)}{2a(n−1)}∫\frac{du}{u^{n-1}\sqrt{a+bu}}
