I am trying to solve the following integral $$\int \frac{\sqrt{(a^2 - x^2)^n}}{x} dx.$$
Can anyone provide a hint for solving the above integral?
Here $a\in\mathbb{R}$ is a real constant and n is an arbitrary natural number.
I'll appreciate any help.
Let $\sqrt{a^2-x^2}=u$
$\implies x\ dx=-u\ du$
$$I_n=\int \frac{\sqrt{(a^2 - x^2)^n}}{x} dx=\int\dfrac{u^{n+1}}{u^2-a^2}du$$
$$I_n-a^2I_{n-2}=\int u^{n-2}du$$