Primitive of Reciprocal of Power of a squared minus x squared

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^n} = \frac x {2 \left({n - 1}\right) a^2 \left({a^2 - x^2}\right)^{n - 1} } + \frac {2 n - 3} {\left({2 n - 2}\right) a^2} \int \frac {\mathrm d x} {\left({a^2 - x^2}\right)^{n - 1} }$

for $x^2 > a^2$.

Proof
Aiming for an expression in the form:
 * $\displaystyle \int u \frac {\mathrm d v} {\mathrm d x} \ \mathrm d x = u v - \int v \ \frac {\mathrm d u} {\mathrm d x} \ \mathrm d x$

in order to use the technique of Integration by Parts, let:

Thus:

Then:

Also see

 * Primitive of $\dfrac 1 {\left({x^2 - a^2}\right)^n}$