Basic Integrals
1. \(\quad \displaystyle ∫u^n\,du=\frac{u^{n+1}}{n+1}+C,\quad n≠−1\)
2. \(\quad \displaystyle ∫\frac{du}{u} =\ln |u|+C\)
3. \(\quad \displaystyle ∫e^u\,du=e^u+C\)
4. \(\quad \displaystyle ∫a^u\,du=\frac{a^u}{\ln a}+C\)
5. \(\quad \displaystyle ∫\sin u\,du=−\cos u+C\)
6. \(\quad \displaystyle ∫\cos u\,du=\sin u+C\)
7. \(\quad \displaystyle ∫\sec^2u\,du=\tan u+C\)
8. \(\quad \displaystyle ∫\csc^2u\,du=−\cot u+C\)
9. \(\quad \displaystyle ∫\sec u\tan u\,du=\sec u+C\)
10. \(\quad \displaystyle ∫\csc u\cot u\,du=−\csc u+C\)
11. \(\quad \displaystyle ∫\tan u\,du=\ln |\sec u|+C\)
12. \(\quad \displaystyle ∫\cot u\,du=\ln |\sin u|+C\)
13. \(\quad \displaystyle ∫\sec u\,du=\ln |\sec u+\tan u|+C\)
14. \(\quad \displaystyle ∫\csc u\,du=\ln |\csc u−\cot u|+C\)
15. \(\quad \displaystyle ∫\frac{du}{\sqrt{a^2−u^2}}=\sin^{−1}\left(\frac{u}{a}\right)+C\)
16. \(\quad \displaystyle ∫\frac{du}{a^2+u^2}=\frac{1}{a}\tan^{−1}\left(\frac{u}{a}\right)+C\)
17. \(\quad \displaystyle ∫\frac{du}{u\sqrt{u^2−a^2}}=\frac{1}{a}\sec^{−1}\frac{|u|}{a}+C\)
Trigonometric Integrals
18. \(\quad \displaystyle ∫\sin^2u\,du=\frac{1}{2}u−\frac{1}{4}\sin 2u+C\)
19. \(\quad \displaystyle ∫\cos^2 u\,du=\frac{1}{2}u+\frac{1}{4}\sin 2u+C\)
20. \(\quad \displaystyle ∫\tan^2 u\,du=\tan u−u+C\)
21. \(\quad \displaystyle ∫\cot ^2 u\,du=−\cot u−u+C\)
22. \(\quad \displaystyle ∫\sin^3 u\,du=−\frac{1}{3}(2+\sin^2u)\cos u+C\)
23. \(\quad \displaystyle ∫\cos^3 u\,du=\frac{1}{3}(2+\cos^2 u)\sin u+C\)
24. \(\quad \displaystyle ∫\tan^3 u\,du=\frac{1}{2}\tan^2 u+\ln |\cos u|+C\)
25. \(\quad \displaystyle ∫\cot^3 u\,du=−\frac{1}{2}\cot^2 u−\ln |\sin u|+C\)
26. \(\quad \displaystyle ∫\sec^3 u\,du=\frac{1}{2}\sec u\tan u+\frac{1}{2}\ln |\sec u+\tan u|+C\)
27. \(\quad \displaystyle ∫\csc^3 u\,du=−\frac{1}{2}\csc u\cot u+\frac{1}{2}\ln |\csc u−\cot u|+C\)
28. \(\quad \displaystyle ∫\sin^n u\,du=\frac{-1}{n}\sin^{n−1}u\cos u+\frac{n−1}{n}∫\sin^{n−2}u\,du\)
29. \(\quad \displaystyle ∫\cos^n u\,du=\frac{1}{n}\cos^{n−1} u\sin u+\frac{n−1}{n}∫\cos^{n−2}u\,du\)
30. \(\quad \displaystyle ∫\tan^n u\,du=\frac{1}{n-1}\tan^{n−1} u−∫\tan^{n−2} u\,du\)
31. \(\quad \displaystyle ∫\cot^n u\,du=\frac{-1}{n-1}\cot^{n−1}u−∫\cot^{n−2}u\,du\)
32. \(\quad \displaystyle ∫\sec^n u\,du=\frac{1}{n-1}\tan u\sec^{n−2}u+\frac{n-2}{n-1}∫\sec^{n−2}u\,du\)
33. \(\quad \displaystyle ∫\csc^n u\,du=\frac{-1}{n-1}\cot u\csc^{n−2}u+\frac{n-2}{n-1}∫\csc^{n−2}u\,du\)
34. \(\quad \displaystyle ∫\sin au\sin bu\,du=\frac{\sin (a−b)u}{2(a−b)}−\frac{\sin (a+b)u}{2(a+b)}+C\)
35. \(\quad \displaystyle ∫\cos au\cos bu\,du=\frac{\sin (a−b)u}{2(a−b)}+\frac{\sin (a+b)u}{2(a+b)}+C\)
36. \(\quad \displaystyle ∫\sin au\cos bu\,du=−\frac{\cos (a−b)u}{2(a−b)}−\frac{\cos (a+b)u}{2(a+b)}+C\)
37. \(\quad \displaystyle ∫u\sin u\,du=\sin u−u\cos u+C\)
38. \(\quad \displaystyle ∫u\cos u\,du=\cos u+u\sin u+C\)
39. \(\quad \displaystyle ∫u^n\sin u\,du=−u^n\cos u+n∫u^{n−1}\cos u\,du\)
40. \(\quad \displaystyle ∫u^n\cos u\,du=u^n\sin u−n∫u^{n−1}\sin u\,du\)
41. \(\quad \begin{align*} \displaystyle ∫\sin^n u\cos^m u\,du = −\frac{\sin^{n−1}u\cos^{m+1}u}{n+m}+\frac{n−1}{n+m}∫\sin^{n−2}u\cos^m u\,du \\[4pt] =\frac{\sin^{n+1}u\cos^{m−1}u}{n+m}+\frac{m−1}{n+m}∫\sin^n u\cos^{m−2}u \,du \end{align*}\)
Exponential and Logarithmic Integrals
42. \(\quad \displaystyle ∫ue^{au}\,du=\frac{1}{a^2}(au−1)e^{au}+C\)
43. \(\quad \displaystyle ∫u^ne^{au}\,du=\frac{1}{a}u^ne^{au}−\frac{n}{a}∫u^{n−1}e^{au}\,du\)
44. \(\quad \displaystyle ∫e^{au}\sin bu\,du=\frac{e^{au}}{a^2+b^2}(a\sin bu−b\cos bu)+C\)
45. \(\quad \displaystyle ∫e^{au}\cos bu\,du=\frac{e^{au}}{a^2+b^2}(a\cos bu+b\sin bu)+C\)
46. \(\quad \displaystyle ∫\ln u\,du=u\ln u−u+C\)
47. \(\quad \displaystyle ∫u^n\ln u\,du=\frac{u^{n+1}}{(n+1)^2}[(n+1)\ln u−1]+C\)
48. \(\quad \displaystyle ∫\frac{1}{u\ln u}\,du=\ln |\ln u|+C\)
Hyperbolic Integrals
49. \(\quad \displaystyle ∫\sinh u\,du=\cosh u+C\)
50. \(\quad \displaystyle ∫\cosh u\,du=\sinh u+C\)
51. \(\quad \displaystyle ∫\tanh u\,du=\ln \cosh u+C\)
52. \(\quad \displaystyle ∫\coth u\,du=\ln |\sinh u|+C\)
53. \(\quad \displaystyle ∫\text{sech}\,u\,du=\tan^{−1}|\sinh u|+C\)
54. \(\quad \displaystyle ∫\text{csch}\,u\,du=\ln ∣\tanh\frac{1}{2}u∣+C\)
55. \(\quad \displaystyle ∫\text{sech}^2 u\,du=\tanh \,u+C\)
56. \(\quad \displaystyle ∫\text{csch}^2 u\,du=−\coth \,u+C\)
57. \(\quad \displaystyle ∫\text{sech} \,u\tanh u\,du=−\text{sech} \,u+C\)
58. \(\quad \displaystyle ∫\text{csch} \,u\coth u\,du=−\text{csch} \,u+C\)
Inverse Trigonometric Integrals
59. \(\quad \displaystyle ∫\sin^{-1}u\,du=u\sin^{-1}u+\sqrt{1−u^2}+C\)
60. \(\quad \displaystyle ∫\cos^{-1}u\,du=u\cos^{-1}u−\sqrt{1−u^2}+C\)
61. \(\quad \displaystyle ∫\tan^{-1}u\,du=u\tan^{-1}u−\frac{1}{2}\ln (1+u^2)+C\)
62. \(\quad \displaystyle ∫u\sin^{-1}u\,du=\frac{2u^2−1}{4}\sin^{-1}u+\frac{u\sqrt{1−u^2}}{4}+C\)
63. \(\quad \displaystyle ∫u\cos^{-1}u\,du=\frac{2u^2−1}{4}\cos^{-1}u-\frac{u\sqrt{1−u^2}}{4}+C\)
64. \(\quad \displaystyle ∫u\tan^{-1}u\,du=\frac{u^2+1}{2}\tan^{-1}u−\frac{u}{2}+C\)
65. \(\quad \displaystyle ∫u^n\sin^{-1}u\,du=\frac{1}{n+1}\left[u^{n+1}\sin^{-1}u−∫\frac{u^{n+1}\,du}{\sqrt{1−u^2}}\right],\quad n≠−1\)
66. \(\quad \displaystyle ∫u^n\cos^{-1}u\,du=\frac{1}{n+1}\left[u^{n+1}\cos^{-1}u+∫\frac{u^{n+1}\,du}{\sqrt{1−u^2}}\right],\quad n≠−1\)
67. \(\quad \displaystyle ∫u^n\tan^{-1}u\,du=\frac{1}{n+1}\left[u^{n+1}\tan^{-1}u−∫\frac{u^{n+1}\,du}{1+u^2}\right],\quad n≠−1\)
Integrals Involving a2 + u2, a > 0
68. \(\quad \displaystyle ∫\sqrt{a^2+u^2}\,du=\frac{u}{2}\sqrt{a^2+u^2}+\frac{a^2}{2}\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
69. \(\quad \displaystyle ∫u^2\sqrt{a^2+u^2}\,du=\frac{u}{8}(a^2+2u^2)\sqrt{a^2+u^2}−\frac{a^4}{8}\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
70. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u}\,du=\sqrt{a^2+u^2}−a\ln \left|\frac{a+\sqrt{a^2+u^2}}{u}\right|+C\)
71. \(\quad \displaystyle ∫\frac{\sqrt{a^2+u^2}}{u^2}\,du=−\frac{\sqrt{a^2+u^2}}{u}+\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
72. \(\quad \displaystyle ∫\frac{du}{\sqrt{a^2+u^2}}=\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
73. \(\quad \displaystyle ∫\frac{u^2}{\sqrt{a^2+u^2}}\,du=\frac{u}{2}\left(\sqrt{a^2+u^2}\right)−\frac{a^2}{2}\ln \left(u+\sqrt{a^2+u^2}\right)+C\)
74. \(\quad \displaystyle ∫\frac{du}{u\sqrt{a^2+u^2}}=\frac{−1}{a}\ln \left|\frac{\sqrt{a^2+u^2}+a}{u}\right|+C\)
75. \(\quad \displaystyle ∫\frac{du}{u^2\sqrt{a^2+u^2}}=−\frac{\sqrt{a^2+u^2}}{a^2u}+C\)
76. \(\quad \displaystyle ∫\frac{du}{\left(a^2+u^2\right)^{3/2}}=\frac{u}{a^2\sqrt{a^2+u^2}}+C\)
Integrals Involving u2 − a2, a > 0
77. \(\quad \displaystyle ∫\sqrt{u^2−a^2}\,du=\frac{u}{2}\sqrt{u^2−a^2}−\frac{a^2}{2}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
78. \(\quad \displaystyle ∫u^2\sqrt{u^2−a^2}\,du=\frac{u}{8}(2u^2−a^2)\sqrt{u^2−a^2}−\frac{a^4}{8}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
79. \(\quad \displaystyle ∫\frac{\sqrt{u^2−a^2}}{u}\,du=\sqrt{u^2−a^2}−a\cos^{-1}\frac{a}{|u|}+C\)
80. \(\quad \displaystyle ∫\frac{\sqrt{u^2−a^2}}{u^2}\,du=−\frac{\sqrt{u^2−a^2}}{u}+\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
81. \(\quad \displaystyle ∫\frac{du}{\sqrt{u^2−a^2}}=\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
82. \(\quad \displaystyle ∫\frac{u^2}{\sqrt{u^2−a^2}}\,du=\frac{u}{2}\sqrt{u^2−a^2}+\frac{a^2}{2}\ln \left|u+\sqrt{u^2−a^2}\right|+C\)
83. \(\quad \displaystyle ∫\frac{du}{u^2\sqrt{u^2−a^2}}=\frac{\sqrt{u^2−a^2}}{a^2u}+C\)
84. \(\quad \displaystyle ∫\frac{du}{(u^2−a^2)^{3/2}}=−\frac{u}{a^2\sqrt{u^2−a^2}}+C\)
Integrals Involving a2 − u2, a > 0
85. \(\quad \displaystyle ∫\sqrt{a^2-u^2}\,du=\frac{u}{2}\sqrt{a^2-u^2}+\frac{a^2}{2}\sin^{-1}\frac{u}{a}+C\)
86. \(\quad \displaystyle ∫u^2\sqrt{a^2-u^2}\,du=\frac{u}{8}(2u^2−a^2)\sqrt{a^2-u^2}+\frac{a^4}{8}\sin^{-1}\frac{u}{a}+C\)
87. \(\quad \displaystyle ∫\frac{\sqrt{a^2-u^2}}{u}\,du=\sqrt{a^2-u^2}−a\ln \left|\frac{a+\sqrt{a^2-u^2}}{u}\right|+C\)
88. \(\quad \displaystyle ∫\frac{\sqrt{a^2-u^2}}{u^2}\,du=\frac{−1}{u}\sqrt{a^2-u^2}−\sin^{-1}\frac{u}{a}+C\)
89. \(\quad \displaystyle ∫\frac{u^2}{\sqrt{a^2-u^2}}\,du=\frac{1}{2}\left(-u\sqrt{a^2-u^2}+a^2\sin^{-1}\frac{u}{a}\right)+C\)
90. \(\quad \displaystyle ∫\frac{du}{u\sqrt{a^2-u^2}}=−\frac{1}{a}\ln \left|\frac{a+\sqrt{a^2-u^2}}{u}\right|+C\)
91. \(\quad \displaystyle ∫\frac{du}{u^2\sqrt{a^2-u^2}}=−\frac{1}{a^2u}\sqrt{a^2-u^2}+C\)
92. \(\quad \displaystyle ∫\left(a^2−u^2\right)^{3/2}\,du=−\frac{u}{8}\left(2u^2−5a^2\right)\sqrt{a^2-u^2}+\frac{3a^4}{8}\sin^{-1}\frac{u}{a}+C\)
93. \(\quad \displaystyle ∫\frac{du}{(a^2−u^2)^{3/2}}=−\frac{u}{a^2\sqrt{a^2−u^2}}+C\)
Integrals Involving 2au − u2, a > 0
94. \(\quad \displaystyle ∫\sqrt{2au−u^2}\,du=\frac{u−a}{2}\sqrt{2au−u^2}+\frac{a^2}{2}\cos^{-1}\left(\frac{a−u}{a}\right)+C\)
95. \(\quad \displaystyle ∫\frac{du}{\sqrt{2au−u^2}}=\cos^{-1}\left(\frac{a−u}{a}\right)+C\)
96. \(\quad \displaystyle ∫u\sqrt{2au−u^2}\,du=\frac{2u^2−au−3a^2}{6}\sqrt{2au−u^2}+\frac{a^3}{2}\cos^{-1}\left(\frac{a−u}{a}\right)+C\)
97. \(\quad \displaystyle ∫\frac{du}{u\sqrt{2au−u^2}}=−\frac{\sqrt{2au−u^2}}{au}+C\)
Integrals Involving a + bu, a ≠ 0
98. \(\quad \displaystyle ∫\frac{u}{a+bu}\,du=\frac{1}{b^2}(a+bu−a\ln |a+bu|)+C\)
99. \(\quad \displaystyle ∫\frac{u^2}{a+bu}\,du=\frac{1}{2b^3}\left[(a+bu)^2−4a(a+bu)+2a^2\ln |a+bu|\right]+C\)
100. \(\quad \displaystyle ∫\frac{du}{u(a+bu)}=\frac{1}{a}\ln \left|\frac{u}{a+bu}\right|+C\)
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. \(\quad \displaystyle ∫\frac{du}{u^n\sqrt{a+bu}}=−\frac{\sqrt{a+bu}}{a(n−1)u^{n−1}}−\frac{b(2n−3)}{2a(n−1)}∫\frac{du}{u^{n-1}\sqrt{a+bu}}\)
