INTEGRALS CONTAINING a²-x²

We assume x²<a²:

1.

$\displaystyle \int\displaystyle \frac{dx}{a^{\displaystyle2}-x^{\displaystyle2}}=\displaystyle \frac{1}{2a}\ln\left(\displaystyle \frac{a+x}{a-x}\right)\;\;$ or $\;\;\displaystyle \frac{1}{a}\tanh^{\displaystyle-1}\displaystyle \frac{x}{a}$

2.

$\displaystyle \int\displaystyle \frac{x\,dx}{a^{\displaystyle2}-x^{\displaystyle2}}=-\displaystyle \frac{1}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})$

3.

$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{a^{\displaystyle2... ...le2}}=-x+\displaystyle \frac{a}{2}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$

4.

$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{a^{\displaystyle2... ...laystyle \frac{a^{\displaystyle2}}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})$

5.

$\displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$

6.

$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(a^{\displaystyle2}... ...tyle \frac{1}{2a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$

7.

$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(a^{\displaystyle2}... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$

8.

$\displaystyle \int\displaystyle \frac{dx}{(a^{\displaystyle2}-x^{\displaystyle2... ...tyle \frac{1}{4a^{\displaystyle3}}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$

9.

$\displaystyle \int\displaystyle \frac{x\,dx}{(a^{\displaystyle2}-x^{\displaysty... ...isplaystyle2}}=\displaystyle \frac{1}{2(a^{\displaystyle2}-x^{\displaystyle2})}$

10.

$\displaystyle \int\displaystyle \frac{x^{\displaystyle2}\,dx}{(a^{\displaystyle... ...yle2})}-\displaystyle \frac{1}{4a}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$

11.

$\displaystyle \int\displaystyle \frac{x^{\displaystyle3}\,dx}{(a^{\displaystyle... ...aystyle2})}+\displaystyle \frac{1}{2}\ln(a^{\displaystyle2}-x^{\displaystyle2})$

12.

$\displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$

13.

$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle2}(a^{\displaystyle2}... ...tyle \frac{3}{4a^{\displaystyle5}}\ln\left(\displaystyle \frac{a+x}{a-x}\right)$

14.

$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle3}(a^{\displaystyle2}... ...aystyle \frac{x^{\displaystyle2}}{a^{\displaystyle2}-x^{\displaystyle2}}\right)$

15.

$\displaystyle \int\displaystyle \frac{dx}{(a^{\displaystyle2}-x^{\displaystyle2... ...laystyle \frac{dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$

16.

$\displaystyle \int\displaystyle \frac{x\,dx}{(a^{\displaystyle2}-x^{\displaysty... ...yle \frac{1}{2(n-1)(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$

17.

$\displaystyle \int\displaystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle... ...aystyle \frac{dx}{x(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$

18.

$\displaystyle \int\displaystyle \frac{x^{\displaystyle m}\,dx}{(a^{\displaystyl... ...laystyle m-2}\,dx}{(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n-1}}$

19.

$\displaystyle \int\displaystyle \frac{dx}{x^{\displaystyle m}(a^{\displaystyle2... ...x^{\displaystyle m-2}(a^{\displaystyle2}-x^{\displaystyle2})^{\displaystyle n}}$

INTEGRALS CONTAINING a2-x2

INTEGRALS CONTAINING a²-x²