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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.

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    See here: http://www.wolframalpha.com/input/?i=integrate+(a%5E2-x%5E2)%5E(n%2F2)%2Fx – 2017-02-17

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Let $\sqrt{a^2-x^2}=u$

$\implies x\ dx=-u\ du$

$$I_n=\int \frac{\sqrt{(a^2 - x^2)^n}}{x} dx=\int\dfrac{u^{n+1}}{u^2-a^2}du$$

$$I_n-a^2I_{n-2}=\int u^{n-2}du$$