Table of Integrals
Table of Integrals
An extended set of Integrals of functions
MST 224 HB pg 26
\(\int a\,dx=ax +c\)
\(\int x^a\,dx\,(a\ne -1)=\frac{x^{a+1}}{a+1} +c\)
\(\int \frac{1}{ax+b}\,dx=\frac{1}{a}\ln|ax+b| +c\)
\(\int e^{ax}\,dx=\frac{1}{a}e^{ax} +c\)
\(\int \ln(ax)\,dx=x(\ln(ax)-1) +c\)
\(\int \sin(ax)\,dx=-\frac{1}{a}\cos(ax) +c\)
\(\int \cos(ax)\,dx=\frac{1}{a}\sin(ax) +c\)
\(\int \tan(ax)\,dx=-\frac{1}{a}\ln|cos(ax)| +c\)
\(\int \cot(ax)\,dx=\frac{1}{a}\ln(|\sin(ax)| +c\)
\(\int \sec(ax)\,dx=\frac{1}{a}\ln|\sec(ax)+\tan(ax)| +c\)
\(\int \csc(ax)\,dx=\frac{1}{a}\ln|\csc(ax)-\cot(ax)| +c\)
\(\int \sec^2(ax)\,dx=\frac{1}{a}\tan(ax) +c\)
\(\int \csc^2(ax)\,dx=-\frac{1}{a}\cot(ax) +c\)
\(\int x\sin(ax)\,dx=\frac{1}{a^2}(\sin(ax)-ax\cos(ax)) +c\)
\(\int x\cos(ax)\,dx=\frac{1}{a^2}(\cos(ax)+ax\sin(ax)) +c\)
\(\int \frac{1}{x^2+a^2}\,dx=\frac{1}{a}\tan^{-1}(\frac{x}{a}) +c\)
\(\int \frac{1}{(x-a)(x-b)}\,dx=\frac{1}{a-b}\ln\bigg|\frac{a-x}{x-b}\bigg| +c\)
\(\int \frac{1}{\sqrt{x^2+a^2}}\,dx=\ln(x+\sqrt{x^2+a^2}) +c\)
\(\int \frac{1}{\sqrt{x^2-a^2}}\,dx=\ln|x+\sqrt{x^2-a^2}| +c\)
\(\int \frac{1}{\sqrt{a^2-x^2}}\,dx=\sin^{-1}\bigg(\frac{x}{a}\bigg) +c\)
Rogawski Calculus
\(\int a^u\,du=\frac{a^u}{\ln a} +C\,\text{ 4.}\)
\(\int \sec u\tan u\,du=\sec u +C\,\text{ 9.}\)
\(\int \csc u\cot u\,du=-\csc u +C\text{ 10.}\)
\(\int \tan u\,du=\ln|\sec u| +C\text{ 11.}\)
\(\int u\,e^{au}\,du=\frac{1}{a^2}(au-1)e^{au} +C\,\text{ 17.}\)
\(\int u^n e^{au}\,du=\frac{1}{a}u^ne^{au}-\frac{n}{a}\int u^{n-1}e^{au}\,du +C\text{ 18.}\)
\(\int e^{au}\sin bu\,du=\frac{e^{au}}{a^2+b^2}(a\sin bu-b\cos bu) +C\text{ 19. }\)
\(\int e^{au}\cos bu\,du=\frac{e^{au}}{a^2+b^2}(a\cos bu+b\sin bu) +C\text{ 20. }\)
\(\int u^n\ln u\,du=\frac{u^{n+1}}{(n+1)^2}[(n+1)\ln u-1]+C\text{ 22. }\)
\(\int \frac{1}{u\ln u}\,du=\ln|\ln u|+C\text{ 23. }\)
\(\int \sinh u\,du=\cosh u+C\text{ 24. }\)
\(\int \cosh u\,du=\sinh u+C\text{ 25. }\)
\(\int \tanh u\,du=\ln\cosh u+C\text{ 26. }\)
\(\int \coth u\,du=\ln|\sinh u|+C\text{ 27. }\)
\(\int sech u\,du=\tan^{-1}|\sinh u|+C\text{ 28. }\)
\(\int csch\,u\,du=\ln\bigg|\tanh\frac{1}{2} u\bigg|+C\text{ 29. }\)
\(\int sech^2 u\,du=\tanh u+C\text{ 30. }\)
\(\int csch^2 u\,du=-\coth u+C\text{ 31. }\)
\(\int sech \,u\tanh u\,du=-sech\,u+C\text{ 32. }\)
\(\int csch\,u\coth u\,du=-csch\,u+C\text{ 33. }\)
\(\int\sin^2u\,du=\tfrac{1}{2}u-\tfrac{1}{4}\sin2u+C\text{ 34. }\)
\(\int\cos^2 u\,du=\tfrac{1}{2}u+\tfrac{1}{4}\sin 2u+C\text{ 35.}\)
\(\int\tan^2 u\,du=\tan u-u+C\text{ 36.}\)
\(\int\cot^2 u\,du=-\cot u-u+C\text{ 37. }\)
\(\int\sin^3 u\,du=-\tfrac{1}{3}(2+\sin^2u)\cos u+C\text{ 38. }\)
\(\int\cos^3 u\, du=\tfrac{1}{3}(2+\cos^2 u)\sin u+C\text{ 39. }\)
\(\int \tan^3 u\,du=\tfrac{1}{2}\tan^2 u+\ln|\cos u|+C\text{ 40. }\)
\(\int\cot^3 u\,du=-\tfrac{1}{2}\cot^2 u-\ln|\sin u|+C\text{ 41. }\)
\(\int\sec^3 u\,du=\tfrac{1}{2}\sec u\tan u+\tfrac{1}{2}\ln|\sec u+\tan u|+C\text{ 42. }\)
\(\int\csc^3 u\,du=-\frac{1}{n}\ln|\csc u|+C\text{ 43. }\)
\(\int\sin^nu\,du=-\frac{1}{n}\sin^{n-1}u\cos u+\frac{n-1}{n}\int\sin^{n-2}u\,du\text{ 44. }\)
\(\int\cos^n u\,du=\frac{1}{n}\cos^{n-1}u\sin u+\frac{n-1}{n}\int\cos^{n-2}u\,du\text{ 45. }\)
\(\int\tan^nu\,du=\frac{1}{n-1}\tan^{n-1}u-\int\tan^{n-2}u\,du\text{ 46.}\)
\(\int\cot^n u\,du=\frac{-1}{n-1}\cot^{n-1}u-\int\cot^{n-2}u\,du\text{ 47. }\)
\(\int\sec^nu\,du=\frac{1}{n-1}\tan u\sec^{n-2}u+\frac{n-2}{n-1}\int\sec^{n-2}u\,du\text{ 48. }\)
\(\int\csc^n u\,du=\frac{-1}{n-1}\cot u\csc^{n-2}u+\frac{n-2}{n-1}\int\csc^{n-2}u \,du\text{ 49. }\)
\(\int\sin au\sin bu\,du=\frac{\sin(a-b)u}{2(a-b)}-\frac{\sin(a+b)u}{2(a+b)}+C\text{ 50. }\)
\(\int\cos au\cos bu\,du=\frac{\sin(a-b)u}{2(a-b)}+\frac{\sin(a+b)u}{2(a+b)}+C\text{ 51. }\)
\(\int\sin au\cos bu\,du=-\frac{\cos(a-b)u}{2(a-b)}-\frac{\cos(a+b)u}{2(a+b)}+C\text{ 52. }\)
\(\int u\sin u\,du=\sin u-u\cos u+C\text{ 53. }\)
\(\int u\cos u\,du=\cos u+u\sin u+C\text{ 54. }\)
\(\int u^n\sin u\,du=-u^n\cos u+n\int u^{n-1}\cos u\,du\text{ 55. }\)
\(\int u^n\cos u\,du=u^n\sin u-n\int u^{n-1}\sin u\,du\text{ 56. }\)
\(\int\sin^nu\cos^mu\,du=-\frac{\sin^{n-1}u\cos^{m+1}u}{n+m}+\frac{n-1}{n+m}\int\sin^{n-2}u\cos^mu\,du=\frac{\sin^{n-1}u\cos^{m+1}u}{n+m}+\frac{m-1}{n+m}\int\sin^nu\cos^{m-2}u\,du\text{ 57. }\)
\(\int\sin^{-1}u\,du=u\sin^{-1}u+\sqrt{1-u^2}+C\text{ 58. }\)
\(\int\cos^{-1}u\,du=u\cos^{-1}u-\sqrt{1-u^2}+C\text{ 59. }\)
\(\int\tan^{-1}u\,du=u\tan^{-1}u-\frac{1}{2}\ln(1+u^2)+C\text{ 60. }\)
\(\int u\sin^{-1}u\,du=\frac{2u^2-1}{4}\sin^{-1}u+\frac{u\sqrt{1-u^2}}{4}+C\text{ 61. }\)
\(\int u\cos^{-1}u\,du=\frac{2u^2-1}{4}\cos^{-1}u-\frac{u\sqrt{1-u^2}}{4}+C\text{ 62. }\)
\(\int u\tan^{-1}u\,du=\frac{u^2+1}{2}\tan^{-1}u-\frac{u}{2}+C\text{ 63. }\)
\(\int u^n\sin^{-1}u\,du=\frac{1}{n+1}\bigg[u^{n+1}\sin^{-1}u-\int\frac{u^{n+1}du}{\sqrt{1-u^2}}\bigg]\,n\ne 1\text{ 64. }\) \(\int u^n\cos^{-1}u\,du=\frac{1}{n+1}\bigg[u^{n+1}\cos^{-1}u+\int\frac{u^{n+1}du}{\sqrt{1-u^2}}\bigg]\,n\ne 1\text{ 65. }\) \(\int u^n\tan^{-1}u\,du=\frac{1}{n+1}\bigg[u^{n+1}\tan^{-1}u-\int\frac{u^{n+1}du}{\sqrt{1+u^2}}\bigg]\,n\ne 1\text{ 66. }\)
\(\int\sqrt{a^2-u^2}\,du=\frac{u}{2}\sqrt{a^2-u^2}+\frac{a^2}{2}\sin^{-1}\frac{u}{a}+C\text{ 67. }\)
\(\int u^2\sqrt{a^2-u^2}\,du=\frac{u}{8}(2u^2-a^2)\sqrt{a^2-u^2}+\frac{a^4}{8}\sin^{-1}\frac{u}{a}+C\text{ 68. }\)
\(\int\frac{\sqrt{a^2-u^2}}{u}\,du=\sqrt{a^2-u^2}-a\ln\bigg|\frac{a+\sqrt{a^2-u^2}}{u}\bigg|+C\text{ 69.}\)
\(\int\frac{\sqrt{a^2-u^2}}{u^2}\,du=-\frac{1}{u}\sqrt{a^2-u^2}-\sin^{-1}\frac{u}{a}+C\text{ 70. }\)
\(\int\frac{u^2\,du}{\sqrt{a^2-u^2}}=-\frac{u}{2}\sqrt{a^2-u^2}+\frac{a^2}{2}\sin^{-1}\frac{u}{a}+C\text{ 71. }\)
\(\int\frac{du}{u\sqrt{a^2-u^2}}=-\frac{1}{a}\ln\bigg|\frac{a+\sqrt{a^2-u^2}}{u}\bigg|+C\text{ 72. }\)
\(\int\frac{du}{u^2\sqrt{a^2-u^2}}=-\frac{1}{a^2u}\sqrt{a^2-u^2}=C\text{ 73.}\)
\(\int(a^2-u^2)^{3/2}\,du=-\frac{u}{8}(2u^2-5a^2)\sqrt{a^2-u^2}+\frac{3a^4}{8}\sin^{-1}\frac{u}{a}+C\text{ 74. }\)
\(\int\frac{du}{(a^2-u^2)^{3/2}}=\frac{u}{a^2\sqrt{a2-u^2}}+C\text{ 75. }\)