INTEGRALS CONTAINING THE SQUARE ROOT OF x²-a²

1.

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

2.

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

3.

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

4.

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

5.

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

6.

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

7.

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

8.

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

9.

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

10.

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

11.

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

12.

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

13.

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

14.

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

15.

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

16.

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

17.

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

18.

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

19.

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

20.

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

21.

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

22.

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

23.

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

24.

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

25.

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

26.

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

27.

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

28.

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

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