Integrals with Inverse Trigonometric Functions

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Integrals with Inverse Trigonometric Functions

1.: $\displaystyle\int\sin^{-1}\displaystyle \frac{x}{a}dx=x\sin^{-1} \displaystyle \frac{x}{a}+\displaystyle \sqrt{a^2-x^2}$
2.: $\displaystyle\int x\sin^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \fr... ...\displaystyle \frac{x}{a}+\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$
3.: $\displaystyle\int x^2\sin^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...yle \frac{x}{a}+\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2)}}{9}$
4.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x}dx=\displaystyle \frac{x... ...rac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdot \cdot \cdot$
5.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$
6.: $\displaystyle\int\left(\sin^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\s... ...a}\right)^2 -2x+2\displaystyle \sqrt{a^2-x^2}\sin^{-1}\displaystyle \frac{x}{a}$
7.: $\displaystyle\int\cos^{-1}\displaystyle \frac{x}{a}dx=x\cos^{-1}\displaystyle \frac{x}{a}-\displaystyle \sqrt{a^2-x^2}$
8.: $\displaystyle\int x\cos^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \fr... ...\displaystyle \frac{x}{a}-\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$

9.: $\displaystyle\int x^2\cos^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...tyle \frac{x}{a}-\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2}}{9}$
10.: $\displaystyle\int\displaystyle \frac{\cos^{-1}(x/a)}{x}dx=\displaystyle \frac{\pi}{2}\ln x-\int\displaystyle \frac{\sin^{-1}(x/a)}{x}dx$
11.: $\displaystyle\int\displaystyle \frac{\cos^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...rac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$
12.: $\displaystyle\int\left(\cos^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\c... ...{a}\right)^2-2x-2\displaystyle \sqrt{a^2-x^2}\cos^{-1}\displaystyle \frac{x}{a}$
13.: $\displaystyle\int\tan^{-1}\displaystyle \frac{x}{a}dx=x\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{a}{2}\ln(x^2+a^2)$
14.: $\displaystyle\int x\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{1}{2}(x^2+a^2)\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{ax}{2}$
15.: $\displaystyle\int x^2\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...frac{x}{a}-\displaystyle \frac{ax^2}{6}+\displaystyle \frac{a^3}{6}\ln(x^2+a^2)$

16.: $\displaystyle\int\displaystyle \frac{\tan^{-1}(x/a)}{x}dx=\displaystyle \frac{x... ...playstyle \frac{(x/a)^5}{5^2}-\displaystyle \frac{(x/a)^7}{7^2}+\cdot\cdot\cdot$
17.: $\displaystyle\int\displaystyle \frac{\tan^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...{a}-\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2+a^2}{x^2}\right)$
18.: $\displaystyle\int\cot^{-1}\displaystyle \frac{x}{a}dx=x\cot^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{a}{2}\ln(x^2+a^2)$
19.: $\displaystyle\int x\cot^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{1}{2}(x^2+a^2)\cot^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{ax}{2}$
20.: $\displaystyle\int x^2\cot^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...frac{x}{a}+\displaystyle \frac{ax^2}{6}-\displaystyle \frac{a^3}{6}\ln(x^2+a^2)$
21.: $\displaystyle\int\displaystyle \frac{\cot^{-1}(x/a)}{x}dx=\displaystyle \frac{\pi}{2}\ln x-\int\displaystyle \frac{\tan^{-1}(x/a)}{x}dx$

22.: $\displaystyle\int\displaystyle \frac{\cot^{-1}(x/a)}{x^2}dx=-\displaystyle \fra... ...{x}+\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2+a^2}{x^2}\right)$
23.: $\displaystyle\int\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{ll... ...style \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
24.: $\displaystyle\int x\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{l... ...ystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
25.: $\displaystyle\int x^2\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}... ...ystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
26.: $\displaystyle\int\displaystyle \frac{\sec^{-1}(x/a)}{x}dx=\displaystyle \frac{\... ...\frac{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdot\cdot\cdot$
27.: $\displaystyle\int\displaystyle \frac{\sec^{-1}(x/a)}{x^2}dx=\left\{ \begin{arra... ...ystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
28.: $\displaystyle\int\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \displaystyle\beg... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$
29.: $\displaystyle\int x\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{l... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$

30.: $\displaystyle\int x^2\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$
31.: $\displaystyle\int\displaystyle \frac{\csc^{-1}(x/a)}{x}dx=-\left(\displaystyle ... ...{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7}+\cdot\cdot\cdot \right)$
32.: $\displaystyle\int\displaystyle \frac{\csc^{-1}(x/a)}{x^2}dx=\left\{ \begin{arra... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$
33.: $\displaystyle\int x^m\sin^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...e \frac{1}{m+1}\int\displaystyle \frac{x^{m+1}}{\displaystyle \sqrt{a^2-x^2}}dx$
34.: $\displaystyle\int x^m\cos^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...e \frac{1}{m+1}\int\displaystyle \frac{x^{m+1}}{\displaystyle \sqrt{a^2-x^2}}dx$
35.: $\displaystyle\int x^m\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...ac{x}{a}-\displaystyle \frac{a}{m+1}\int\displaystyle \frac{x^{m+1}}{x^2+a^2}dx$
36.: $\displaystyle\int x^m\cot^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x... ...ac{x}{a}+\displaystyle \frac{a}{m+1}\int\displaystyle \frac{x^{m+1}}{x^2+a^2}dx$
37.: $\displaystyle\int x^m\sec^{-1}\displaystyle \frac{x}{a}=\left \{ \begin{array}{... ...aystyle \frac{\pi}{2}<sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array}\right.$
38.: $\displaystyle\int x^m\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$

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Integrals with Inverse Trigonometric Functions