INTEGRALS CONTAINING THE SQUARE ROOT OF x2+a2

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

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.
$\displaystyle \int\displaystyle \frac{dx}{x\displaystyle \sqrt{x^{\displaystyle... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

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

7.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}\displaystyle \sqrt... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

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.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

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

14.
$\displaystyle \int\displaystyle \frac{\displaystyle \sqrt{x^{\displaystyle2}+a^... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

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.
$\displaystyle \int\displaystyle \frac{dx}{x(x^{\displaystyle2}+a^{\displaystyle... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

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

21.
$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(x^{\displaystyle2}... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

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

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

26.
$\displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

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

28.
$\displaystyle \int\displaystyle \frac{(x^{\displaystyle2}+a^{\displaystyle2})^{... ...e \frac{a+\displaystyle \sqrt{x^{\displaystyle2}+a^{\displaystyle2}}}{x}\right)$

[Tables]

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