Matematica

Integrali con funzioni trigonometriche inverse

1.: $\displaystyle\int\sin^{-1}\displaystyle \frac{x}{a}dx=x\sin^{-1} \displaystyle \frac{x}{a}+\displaystyle \sqrt{a^2-x^2}$

2.: $\displaystyle\int x\sin^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \frac{x^2}{2}-\displaystyle \frac{a^2}{4}\right)\sin^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$

3.: $\displaystyle\int x^2\sin^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x^3}{3}\sin^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2)}}{9}$

4.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x}dx=\displaystyle \frac{x}{a}+\displaystyle \frac{(x/a)^3}{2\cdot 3\cdot 3}+\displaystyle \frac{1\cdot 3(x/a)^5}{2 \cdot 4\cdot 5\cdot 5\cdot}+\displaystyle \frac{1\cdot 3\cdot 5(x/a)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdot \cdot \cdot$

5.: $\displaystyle\int\displaystyle \frac{\sin^{-1}(x/a)}{x^2}dx=-\displaystyle \frac{\sin^{-1}(x/a)}{x}-\displaystyle \frac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$

6.: $\displaystyle\int\left(\sin^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\sin^{-1}\displaystyle \frac{x}{a}\right)^2 -2x+2\displaystyle \sqrt{a^2-x^2}\sin^{-1}\displaystyle \frac{x}{a}$

7.: $\displaystyle\int\cos^{-1}\displaystyle \frac{x}{a}dx=x\cos^{-1}\displaystyle \frac{x}{a}-\displaystyle \sqrt{a^2-x^2}$

8.: $\displaystyle\int x\cos^{-1}\displaystyle \frac{x}{a}dx=\left(\displaystyle \frac{x^2}{2}-\displaystyle \frac{a^2}{4}\right)\cos^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{x\displaystyle \sqrt{a^2-x^2}}{4}$

9.: $\displaystyle\int x^2\cos^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x^3}{3}\cos^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{(x^2+2a^2)\displaystyle \sqrt{a^2-x^2}}{9}$

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.: $\displaystyle\int\displaystyle \frac{\cos^{-1}(x/a)}{x^2}dx=-\displaystyle \frac{\cos^{-1}(x/a)}{x}+\displaystyle \frac{1}{a}\ln\left(\displaystyle \frac{a+\displaystyle \sqrt{a^2-x^2}}{x}\right)$

12.: $\displaystyle\int\left(\cos^{-1}\displaystyle \frac{x}{a}\right)^2 dx=x\left(\cos^{-1}\displaystyle \frac{x}{a}\right)^2-2x-2\displaystyle \sqrt{a^2-x^2}\cos^{-1}\displaystyle \frac{x}{a}$

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.: $\displaystyle\int x^2\tan^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x^3}{3}\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{ax^2}{6}+\displaystyle \frac{a^3}{6}\ln(x^2+a^2)$

16.: $\displaystyle\int\displaystyle \frac{\tan^{-1}(x/a)}{x}dx=\displaystyle \frac{x}{a}-\displaystyle \frac{(x/a)^3}{3^2}+\displaystyle \frac{(x/a)^5}{5^2}-\displaystyle \frac{(x/a)^7}{7^2}+\cdot\cdot\cdot$

17.: $\displaystyle\int\displaystyle \frac{\tan^{-1}(x/a)}{x^2}dx=-\displaystyle \frac{1}{x}\tan^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2+a^2}{x^2}\right)$

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.: $\displaystyle\int x^2\cot^{-1}\displaystyle \frac{x}{a}dx=\displaystyle \frac{x^3}{3}\cot^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{ax^2}{6}-\displaystyle \frac{a^3}{6}\ln(x^2+a^2)$

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.: $\displaystyle\int\displaystyle \frac{\cot^{-1}(x/a)}{x^2}dx=-\displaystyle \frac{\cot^{-1}(x/a)}{x}+\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{x^2+a^2}{x^2}\right)$

23.: $\displaystyle\int\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{ll} x\sec^{-1}\displaystyle \frac{x}{a}-a\ln(x+\displaystyle \sqrt{x^2-a^2}) & 0<\sec^{-1}\displaystyle \frac{x}{a}<\displaystyle \frac{\pi}{2}\\ \\ x\sec^{-1}\displaystyle \frac{x}{a}+a\ln(x+\displaystyle \sqrt{x^2-a^2})& \displaystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array} \right.$

24.: $\displaystyle\int x\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{ll} \displaystyle \frac{x^2}{2}\sec^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{a\displaystyle \sqrt{x^2-a^2}}{2} & 0<\sec^{-1}\displaystyle \frac{x}{a}<\displaystyle \frac{\pi}{2}\\ \\ \displaystyle \frac{x^2}{2}\sec^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{a\displaystyle \sqrt{x^2-a^2}}{2}& \displaystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array} \right.$

25.: $\displaystyle\int x^2\sec^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{ll} \displaystyle \frac{x^3}{3}\sec^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{ax\displaystyle \sqrt{x^2-a^2}}{6}-\displaystyle \frac{a^3}{6}\ln(x+\displaystyle \sqrt{x^2-a^2}) & 0<\sec^{-1}\displaystyle \frac{x}{a}<\displaystyle \frac{\pi}{2}\\ \\ \displaystyle \frac{x^3}{3}\sec^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{ax\displaystyle \sqrt{x^2-a^2}}{6}-\displaystyle \frac{a^3}{6}\ln(x+\displaystyle \sqrt{x^2-a^2}) & \displaystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array} \right.$

26.: $\displaystyle\int\displaystyle \frac{\sec^{-1}(x/a)}{x}dx=\displaystyle \frac{\pi}{2}\ln x+\displaystyle \frac{a}{x}+\displaystyle \frac{(a/x)^3}{2\cdot 3\cdot 3}+\displaystyle \frac{1\cdot 3(a/x)^5}{2\cdot 4\cdot 5\cdot 5} + \displaystyle \frac{1\cdot 3\cdot 5(a/x)^7}{2\cdot 4\cdot 6\cdot 7\cdot 7} + \cdot\cdot\cdot$

27.: $\displaystyle\int\displaystyle \frac{\sec^{-1}(x/a)}{x^2}dx=\left\{ \begin{array}{ll} -\displaystyle \frac{\sec^{-1}(x/a)}{x}+\displaystyle \frac{\displaystyle \sqrt{x^2-a^2}}{ax} & 0<\sec^{-1}\displaystyle \frac{x}{a}<\displaystyle \frac{\pi}{2}\\ \\ -\displaystyle \frac{\sec^{-1}(x/a)}{x}-\displaystyle \frac{\displaystyle \sqrt{x^2-a^2}}{ax} & \displaystyle \frac{\pi}{2}<\sec^{-1}\displaystyle \frac{x}{a}<\pi \end{array} \right.$

28.: $\displaystyle\int\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \displaystyle\begin{array}{ll} \displaystyle x\csc^{-1}\displaystyle \frac{x}{a}+a\ln(x+\displaystyle \sqrt{x^2-a^2}) & 0<\csc^{-1}\displaystyle \frac{x}{a}<\displaystyle \frac{\pi}{2}\\ \\ \displaystyle x\csc^{-1}\displaystyle \frac{x}{a}-a\ln(x+\displaystyle \sqrt{x^2-a^2}) & -\displaystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array} \right.$

29.: $\displaystyle\int x\csc^{-1}\displaystyle \frac{x}{a}dx=\left\{ \begin{array}{ll} \displaystyle \frac{x^2}{2}\csc^{-1}\displaystyle \frac{x}{a}+\displaystyle \frac{a\displaystyle \sqrt{x^2-a^2}}{2} & 0<\csc^{-1}\displaystyle \frac{x}{a}<\displaystyle \frac{\pi}{2}\\ \\ \displaystyle \frac{x^2}{2}\csc^{-1}\displaystyle \frac{x}{a}-\displaystyle \frac{a\displaystyle \sqrt{x^2-a^2}}{2} & -\displaystyle \frac{\pi}{2}<\csc^{-1}\displaystyle \frac{x}{a}<0 \end{array} \right.$