INTEGRALS THAT CONTAIN TRIGONOMETRIC RECIPROCALS

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.

$\displaystyle\int x^m\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}... ...laystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array}\right.$