INTEGRALS CONTAINING THE SQUARE ROOT OF x2-a2

1.
$\displaystyle \int\displaystyle \frac{dx}{\displaystyle \sqrt{x^{\displaystyle2...
...}}=\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

2.
$\displaystyle \int\displaystyle \frac{x\,dx}{\displaystyle \sqrt{x^{\displaysty...
...a^{\displaystyle2}}}=\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}$

3.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{\displaystyle \sq...
...{2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

4.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{\displaystyle \sq...
...3}+a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}$

5.
$\displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{x^{\displaystyle...
...\frac{1}{a}\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

6.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}\displaystyle \sqrt...
...displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}}{a^{\displaystyle2}x}$

7.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}\displaystyle \sqrt...
...laystyle3}}\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

8.
$\displaystyle \int\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\,d...
...{2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

9.
$\displaystyle \int x\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\...
...splaystyle \frac{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle3/2}}{3}$

10.
$\displaystyle \int x^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}-a^{...
...{8}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

11.
$\displaystyle \int x^{\displaystyle3}\displaystyle \sqrt{x^{\displaystyle2}-a^{...
...^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle3/2}}{3}$

12.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}-a^...
...ystyle2}}-a\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

13.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}-a^...
...+\;\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

14.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}-a^...
...frac{1}{2a}\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

15.
$\displaystyle \int\displaystyle \frac{dx}{(x^{\displaystyle2}-a^{\displaystyle2...
...}{a^{\displaystyle2}\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}}$

16.
$\displaystyle \int\displaystyle \frac{x\,dx}{(x^{\displaystyle2}-a^{\displaysty...
...playstyle \frac{-1}{\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}}$

17.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(x^{\displaystyle...
...}}+\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

18.
$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(x^{\displaystyle...
...a^{\displaystyle2}}{\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}}$

19.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}-a^{\displaystyle...
...laystyle3}}\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

20.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(x^{\displaystyle2}...
...}{a^{\displaystyle4}\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}}$

21.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}...
...laystyle5}}\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

22.
$\displaystyle \int(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle3/2}\,d...
...e4}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

23.
$\displaystyle \int x(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle3/2}\...
...splaystyle \frac{(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle5/2}}{5}$

24.
$\begin{array}{ll}
\displaystyle \int x^{\displaystyle2}(x^{\displaystyle2}-a^{\...
...x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)
\end{array}$

25.
$\displaystyle \int x^{\displaystyle3}(x^{\displaystyle2}-a^{\displaystyle2})^{\...
...^{\displaystyle2}(x^{\displaystyle2}-a^{\displaystyle2})^{\displaystyle5/2}}{5}$

26.
$\displaystyle \int\displaystyle \frac{(x^{\displaystyle2}-a^{\displaystyle2})^{...
...playstyle3}\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

27.
$\displaystyle \int\displaystyle \frac{(x^{\displaystyle2}-a^{\displaystyle2})^{...
...e2}\ln\left(x+\displaystyle \sqrt{x^{\displaystyle2}-a^{\displaystyle2}}\right)$

28.
$\displaystyle \int\displaystyle \frac{(x^{\displaystyle2}-a^{\displaystyle2})^{...
...frac{3}{2}a\sec^{\displaystyle-1}\left\vert\displaystyle \frac{x}{a}\right\vert$

[Tables]

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