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- Gilbert Strang & Edwin “Jed” Herman
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Basic Integrals
1. \(\quad \displaystyle \int u^n\,du=\dfrac{u^{n+1}}{n+1}+C,\quad n \neq −1\)
2. \(\quad \displaystyle \int \dfrac{du}{u} =\ln |u|+C\)
3. \(\quad \displaystyle \int e^u\,du=e^u+C\)
4. \(\quad \displaystyle \int a^u\,du=\dfrac{a^u}{\ln a}+C\)
5. \(\quad \displaystyle \int \sin u\,du=−\cos u+C\)
6. \(\quad \displaystyle \int \cos u\,du=\sin u+C\)
7. \(\quad \displaystyle \int \sec^2u\,du=\tan u+C\)
8. \(\quad \displaystyle \int \csc^2u\,du=−\cot u+C\)
9. \(\quad \displaystyle \int \sec u\tan u\,du=\sec u+C\)
10. \(\quad \displaystyle \int \csc u\cot u\,du=−\csc u+C\)
11. \(\quad \displaystyle \int \tan u\,du=\ln |\sec u|+C\)
12. \(\quad \displaystyle \int \cot u\,du=\ln |\sin u|+C\)
13. \(\quad \displaystyle \int \sec u\,du=\ln |\sec u+\tan u|+C\)
14. \(\quad \displaystyle \int \csc u\,du=\ln |\csc u−\cot u|+C\)
15. \(\quad \displaystyle \int \dfrac{du}{\sqrt{a^2−u^2}}=\sin^{−1}\left(\dfrac{u}{a}\right)+C\)
16. \(\quad \displaystyle \int \dfrac{du}{a^2+u^2}=\dfrac{1}{a}\tan^{−1}\left(\dfrac{u}{a}\right)+C\)
17. \(\quad \displaystyle \int \dfrac{du}{u\sqrt{u^2−a^2}}=\dfrac{1}{a}\sec^{−1}\dfrac{|u|}{a}+C\)
Trigonometric Integrals
18. \(\quad \displaystyle \int \sin^2u\,du=\dfrac{1}{2}u−\dfrac{1}{4}\sin 2u+C\)
19. \(\quad \displaystyle \int \cos^2 u\,du=\dfrac{1}{2}u+\dfrac{1}{4}\sin 2u+C\)
20. \(\quad \displaystyle \int \tan^2 u\,du=\tan u−u+C\)
21. \(\quad \displaystyle \int \cot ^2 u\,du=−\cot u−u+C\)
22. \(\quad \displaystyle \int \sin^3 u\,du=−\dfrac{1}{3}(2+\sin^2u)\cos u+C\)
23. \(\quad \displaystyle \int \cos^3 u\,du=\dfrac{1}{3}(2+\cos^2 u)\sin u+C\)
24. \(\quad \displaystyle \int \tan^3 u\,du=\dfrac{1}{2}\tan^2 u+\ln |\cos u|+C\)
25. \(\quad \displaystyle \int \cot^3 u\,du=−\dfrac{1}{2}\cot^2 u−\ln |\sin u|+C\)
26. \(\quad \displaystyle \int \sec^3 u\,du=\dfrac{1}{2}\sec u\tan u+\dfrac{1}{2}\ln |\sec u+\tan u|+C\)
27. \(\quad \displaystyle \int \csc^3 u\,du=−\dfrac{1}{2}\csc u\cot u+\dfrac{1}{2}\ln |\csc u−\cot u|+C\)
28. \(\quad \displaystyle \int \sin^n u\,du=\dfrac{-1}{n}\sin^{n−1}u\cos u+\dfrac{n−1}{n} \int \sin^{n−2}u\,du\)
29. \(\quad \displaystyle \int \cos^n u\,du=\dfrac{1}{n}\cos^{n−1} u\sin u+\dfrac{n−1}{n} \int \cos^{n−2}u\,du\)
30. \(\quad \displaystyle \int \tan^n u\,du=\dfrac{1}{n-1}\tan^{n−1} u− \int \tan^{n−2} u\,du\)
31. \(\quad \displaystyle \int \cot^n u\,du=\dfrac{-1}{n-1}\cot^{n−1}u− \int \cot^{n−2}u\,du\)
32. \(\quad \displaystyle \int \sec^n u\,du=\dfrac{1}{n-1}\tan u\sec^{n−2}u+\dfrac{n-2}{n-1} \int \sec^{n−2}u\,du\)
33. \(\quad \displaystyle \int \csc^n u\,du=\dfrac{-1}{n-1}\cot u\csc^{n−2}u+\dfrac{n-2}{n-1} \int \csc^{n−2}u\,du\)
34. \(\quad \displaystyle \int \sin au\sin bu\,du=\dfrac{\sin (a−b)u}{2(a−b)}−\dfrac{\sin (a+b)u}{2(a+b)}+C\)
35. \(\quad \displaystyle \int \cos au\cos bu\,du=\dfrac{\sin (a−b)u}{2(a−b)}+\dfrac{\sin (a+b)u}{2(a+b)}+C\)
36. \(\quad \displaystyle \int \sin au\cos bu\,du=−\dfrac{\cos (a−b)u}{2(a−b)}−\dfrac{\cos (a+b)u}{2(a+b)}+C\)
37. \(\quad \displaystyle \int u\sin u\,du=\sin u−u\cos u+C\)
38. \(\quad \displaystyle \int u\cos u\,du=\cos u+u\sin u+C\)
39. \(\quad \displaystyle \int u^n\sin u\,du=−u^n\cos u+n \int u^{n−1}\cos u\,du\)
40. \(\quad \displaystyle \int u^n\cos u\,du=u^n\sin u−n \int u^{n−1}\sin u\,du\)
41. \( \quad \begin{array}{rcl}
\displaystyle \int \sin^n u\cos^m u\,du & = & −\dfrac{\sin^{n−1}u\cos^{m+1}u}{n+m}+\dfrac{n−1}{n+m} \int \sin^{n−2}u\cos^m u\,du \\
& = & \dfrac{\sin^{n+1}u\cos^{m−1}u}{n+m}+\dfrac{m−1}{n+m} \int \sin^n u\cos^{m−2}u \,du \\
\end{array} \)
Exponential and Logarithmic Integrals
42. \(\quad \displaystyle \int ue^{au}\,du=\dfrac{1}{a^2}(au−1)e^{au}+C\)
43. \(\quad \displaystyle \int u^ne^{au}\,du=\dfrac{1}{a}u^ne^{au}−\dfrac{n}{a} \int u^{n−1}e^{au}\,du\)
44. \(\quad \displaystyle \int e^{au}\sin bu\,du=\dfrac{e^{au}}{a^2+b^2}(a\sin bu−b\cos bu)+C\)
45. \(\quad \displaystyle \int e^{au}\cos bu\,du=\dfrac{e^{au}}{a^2+b^2}(a\cos bu+b\sin bu)+C\)
46. \(\quad \displaystyle \int \ln u\,du=u\ln u−u+C\)
47. \(\quad \displaystyle \int u^n\ln u\,du=\dfrac{u^{n+1}}{(n+1)^2}[(n+1)\ln u−1]+C\)
48. \(\quad \displaystyle \int \dfrac{1}{u\ln u}\,du=\ln |\ln u|+C\)
Hyperbolic Integrals
49. \(\quad \displaystyle \int \sinh u\,du=\cosh u+C\)
50. \(\quad \displaystyle \int \cosh u\,du=\sinh u+C\)
51. \(\quad \displaystyle \int \tanh u\,du=\ln \cosh u+C\)
52. \(\quad \displaystyle \int \coth u\,du=\ln |\sinh u|+C\)
53. \(\quad \displaystyle \int \text{sech}\,u\,du=\tan^{−1}|\sinh u|+C\)
54. \(\quad \displaystyle \int \text{csch}\,u\,du=\ln ∣\tanh\dfrac{1}{2}u∣+C\)
55. \(\quad \displaystyle \int \text{sech}^2 u\,du=\tanh \,u+C\)
56. \(\quad \displaystyle \int \text{csch}^2 u\,du=−\coth \,u+C\)
57. \(\quad \displaystyle \int \text{sech} \,u\tanh u\,du=−\text{sech} \,u+C\)
58. \(\quad \displaystyle \int \text{csch} \,u\coth u\,du=−\text{csch} \,u+C\)
Inverse Trigonometric Integrals
59. \(\quad \displaystyle \int \sin^{-1}u\,du=u\sin^{-1}u+\sqrt{1−u^2}+C\)
60. \(\quad \displaystyle \int \cos^{-1}u\,du=u\cos^{-1}u−\sqrt{1−u^2}+C\)
61. \(\quad \displaystyle \int \tan^{-1}u\,du=u\tan^{-1}u−\dfrac{1}{2}\ln (1+u^2)+C\)
62. \(\quad \displaystyle \int u\sin^{-1}u\,du=\dfrac{2u^2−1}{4}\sin^{-1}u+\dfrac{u\sqrt{1−u^2}}{4}+C\)
63. \(\quad \displaystyle \int u\cos^{-1}u\,du=\dfrac{2u^2−1}{4}\cos^{-1}u-\dfrac{u\sqrt{1−u^2}}{4}+C\)
64. \(\quad \displaystyle \int u\tan^{-1}u\,du=\dfrac{u^2+1}{2}\tan^{-1}u−\dfrac{u}{2}+C\)
65. \(\quad \displaystyle \int u^n\sin^{-1}u\,du=\dfrac{1}{n+1}\left[u^{n+1}\sin^{-1}u− \int \dfrac{u^{n+1}\,du}{\sqrt{1−u^2}}\right],\quad n \neq −1\)
66. \(\quad \displaystyle \int u^n\cos^{-1}u\,du=\dfrac{1}{n+1}\left[u^{n+1}\cos^{-1}u+ \int \dfrac{u^{n+1}\,du}{\sqrt{1−u^2}}\right],\quad n \neq −1\)
67. \(\quad \displaystyle \int u^n\tan^{-1}u\,du=\dfrac{1}{n+1}\left[u^{n+1}\tan^{-1}u− \int \dfrac{u^{n+1}\,du}{1+u^2}\right],\quad n \neq −1\)
Integrals Involving \(a^2 + u^2\), \(a \gt 0\)
68. \(\quad \displaystyle \int \sqrt{a^2+u^2}\,du=\dfrac{u}{2}\sqrt{a^2+u^2}+\dfrac{a^2}{2}\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
69. \(\quad \displaystyle \int u^2\sqrt{a^2+u^2}\,du=\dfrac{u}{8}(a^2+2u^2)\sqrt{a^2+u^2}−\dfrac{a^4}{8}\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
70. \(\quad \displaystyle \int \dfrac{\sqrt{a^2+u^2}}{u}\,du=\sqrt{a^2+u^2}−a\ln \left|\dfrac{a+\sqrt{a^2+u^2}}{u}\right|+C\)
71. \(\quad \displaystyle \int \dfrac{\sqrt{a^2+u^2}}{u^2}\,du=−\dfrac{\sqrt{a^2+u^2}}{u}+\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
72. \(\quad \displaystyle \int \dfrac{du}{\sqrt{a^2+u^2}}=\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
73. \(\quad \displaystyle \int \dfrac{u^2}{\sqrt{a^2+u^2}}\,du=\dfrac{u}{2}\left(\sqrt{a^2+u^2}\right)−\dfrac{a^2}{2}\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
74. \(\quad \displaystyle \int \dfrac{du}{u\sqrt{a^2+u^2}}=\dfrac{−1}{a}\ln \left|\dfrac{\sqrt{a^2+u^2}+a}{u}\right|+C\)
75. \(\quad \displaystyle \int \dfrac{du}{u^2\sqrt{a^2+u^2}}=−\dfrac{\sqrt{a^2+u^2}}{a^2u}+C\)
76. \(\quad \displaystyle \int \dfrac{du}{\left(a^2+u^2\right)^{3/2}}=\dfrac{u}{a^2\sqrt{a^2+u^2}}+C\)
Integrals Involving \(u^2 - a^2\), \(a \gt 0\)
77. \(\quad \displaystyle \int \sqrt{u^2−a^2}\,du=\dfrac{u}{2}\sqrt{u^2−a^2}−\dfrac{a^2}{2}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
78. \(\quad \displaystyle \int u^2\sqrt{u^2−a^2}\,du=\dfrac{u}{8}(2u^2−a^2)\sqrt{u^2−a^2}−\dfrac{a^4}{8}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
79. \(\quad \displaystyle \int \dfrac{\sqrt{u^2−a^2}}{u}\,du=\sqrt{u^2−a^2}−a\cos^{-1}\dfrac{a}{|u|}+C\)
80. \(\quad \displaystyle \int \dfrac{\sqrt{u^2−a^2}}{u^2}\,du=−\dfrac{\sqrt{u^2−a^2}}{u}+\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
81. \(\quad \displaystyle \int \dfrac{du}{\sqrt{u^2−a^2}}=\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
82. \(\quad \displaystyle \int \dfrac{u^2}{\sqrt{u^2−a^2}}\,du=\dfrac{u}{2}\sqrt{u^2−a^2}+\dfrac{a^2}{2}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
83. \(\quad \displaystyle \int \dfrac{du}{u^2\sqrt{u^2−a^2}}=\dfrac{\sqrt{u^2−a^2}}{a^2u}+C\)
84. \(\quad \displaystyle \int \dfrac{du}{(u^2−a^2)^{3/2}}=−\dfrac{u}{a^2\sqrt{u^2−a^2}}+C\)
Integrals Involving \(a^2 - u^2\), \(a \gt 0\)
85. \(\quad \displaystyle \int \sqrt{a^2-u^2}\,du=\dfrac{u}{2}\sqrt{a^2-u^2}+\dfrac{a^2}{2}\sin^{-1}\dfrac{u}{a}+C\)
86. \(\quad \displaystyle \int u^2\sqrt{a^2-u^2}\,du=\dfrac{u}{8}(2u^2−a^2)\sqrt{a^2-u^2}+\dfrac{a^4}{8}\sin^{-1}\dfrac{u}{a}+C\)
87. \(\quad \displaystyle \int \dfrac{\sqrt{a^2-u^2}}{u}\,du=\sqrt{a^2-u^2}−a\ln \left|\dfrac{a+\sqrt{a^2-u^2}}{u}\right|+C\)
88. \(\quad \displaystyle \int \dfrac{\sqrt{a^2-u^2}}{u^2}\,du=\dfrac{−1}{u}\sqrt{a^2-u^2}−\sin^{-1}\dfrac{u}{a}+C\)
89. \(\quad \displaystyle \int \dfrac{u^2}{\sqrt{a^2-u^2}}\,du=\dfrac{1}{2}\left(-u\sqrt{a^2-u^2}+a^2\sin^{-1}\dfrac{u}{a}\right)+C\)
90. \(\quad \displaystyle \int \dfrac{du}{u\sqrt{a^2-u^2}}=−\dfrac{1}{a}\ln \left|\dfrac{a+\sqrt{a^2-u^2}}{u}\right|+C\)
91. \(\quad \displaystyle \int \dfrac{du}{u^2\sqrt{a^2-u^2}}=−\dfrac{1}{a^2u}\sqrt{a^2-u^2}+C\)
92. \(\quad \displaystyle \int \left(a^2−u^2\right)^{3/2}\,du=−\dfrac{u}{8}\left(2u^2−5a^2\right)\sqrt{a^2-u^2}+\dfrac{3a^4}{8}\sin^{-1}\dfrac{u}{a}+C\)
93. \(\quad \displaystyle \int \dfrac{du}{(a^2−u^2)^{3/2}}=−\dfrac{u}{a^2\sqrt{a^2−u^2}}+C\)
Integrals Involving \(2au - u^2\), \(a \gt 0\)
94. \(\quad \displaystyle \int \sqrt{2au−u^2}\,du=\dfrac{u−a}{2}\sqrt{2au−u^2}+\dfrac{a^2}{2}\cos^{-1}\left(\dfrac{a−u}{a}\right)+C\)
95. \(\quad \displaystyle \int \dfrac{du}{\sqrt{2au−u^2}}=\cos^{-1}\left(\dfrac{a−u}{a}\right)+C\)
96. \(\quad \displaystyle \int u\sqrt{2au−u^2}\,du=\dfrac{2u^2−au−3a^2}{6}\sqrt{2au−u^2}+\dfrac{a^3}{2}\cos^{-1}\left(\dfrac{a−u}{a}\right)+C\)
97. \(\quad \displaystyle \int \dfrac{du}{u\sqrt{2au−u^2}}=−\dfrac{\sqrt{2au−u^2}}{au}+C\)
Integrals Involving \(a + bu\), \(a \neq 0\)
98. \(\quad \displaystyle \int \dfrac{u}{a+bu}\,du=\dfrac{1}{b^2}(a+bu−a\ln |a+bu|)+C\)
99. \(\quad \displaystyle \int \dfrac{u^2}{a+bu}\,du=\dfrac{1}{2b^3}\left[(a+bu)^2−4a(a+bu)+2a^2\ln |a+bu|\right]+C\)
100. \(\quad \displaystyle \int \dfrac{du}{u(a+bu)}=\dfrac{1}{a}\ln \left|\dfrac{u}{a+bu}\right|+C\)
101. \(\quad \displaystyle \int \dfrac{du}{u^2(a+bu)}=−\dfrac{1}{au}+\dfrac{b}{a^2}\ln \left|\dfrac{a+bu}{u}\right|+C\)
102. \(\quad \displaystyle \int \dfrac{u}{(a+bu)^2}\,du=\dfrac{a}{b^2(a+bu)}+\dfrac{1}{b^2}\ln |a+bu|+C\)
103. \(\quad \displaystyle \int \dfrac{u}{u(a+bu)^2}\,du=\dfrac{1}{a(a+bu)}−\dfrac{1}{a^2}\ln \left|\dfrac{a+bu}{u}\right|+C\)
104. \(\quad \displaystyle \int \dfrac{u^2}{(a+bu)^2}\,du=\dfrac{1}{b^3}\left(a+bu−\dfrac{a^2}{a+bu}−2a\ln |a+bu|\right)+C\)
105. \(\quad \displaystyle \int u\sqrt{a+bu}\,du=\dfrac{2}{15b^2}(3bu−2a)(a+bu)^{3/2}+C\)
106. \(\quad \displaystyle \int \dfrac{u}{\sqrt{a+bu}}\,du=\dfrac{2}{3b^2}(bu−2a)\sqrt{a+bu}+C\)
107. \(\quad \displaystyle \int \dfrac{u^2}{\sqrt{a+bu}}\,du=\dfrac{2}{15b^3}(8a^2+3b^2u^2−4abu)\sqrt{a+bu}+C\)
108. \(\quad \displaystyle \int \dfrac{du}{u\sqrt{a+bu}}=\begin{cases} \dfrac{1}{\sqrt{a}}\ln \left|\dfrac{\sqrt{a+bu}−\sqrt{a}}{\sqrt{a+bu}+\sqrt{a}}\right|+C,\quad \text{if}\,a>0\\[4pt] \dfrac{\sqrt{2}}{\sqrt{−a}}\tan^{-1}\sqrt{\dfrac{a+bu}{−a}}+C,\quad \text{if}\,a<0 \end{cases}\)
109. \(\quad \displaystyle \int \dfrac{\sqrt{a+bu}}{u}\,du=2\sqrt{a+bu}+a \int \dfrac{du}{u\sqrt{a+bu}}\)
110. \(\quad \displaystyle \int \dfrac{\sqrt{a+bu}}{u^2}\,du=−\dfrac{\sqrt{a+bu}}{u}+\dfrac{b}{2} \int \dfrac{du}{u\sqrt{a+bu}}\)
111. \(\quad \displaystyle \int u^n\sqrt{a+bu}\,du=\dfrac{2}{b(2n+3)}\left[u^n(a+bu)^{3/2}−na \int u^{n−1}\sqrt{a+bu}\,du\right]\)
112. \(\quad \displaystyle \int \dfrac{u^n}{\sqrt{a+bu}}\,du=\dfrac{2u^n\sqrt{a+bu}}{b(2n+1)}−\dfrac{2na}{b(2n+1)} \int \dfrac{u^{n−1}}{\sqrt{a+bu}}\,du\)
113. \(\quad \displaystyle \int \dfrac{du}{u^n\sqrt{a+bu}}=−\dfrac{\sqrt{a+bu}}{a(n−1)u^{n−1}}−\dfrac{b(2n−3)}{2a(n−1)} \int \dfrac{du}{u^{n-1}\sqrt{a+bu}}\)
