Primitive of Reciprocal of x by x squared plus a squared/Proof 3

Theorem

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

Proof
From Primitive of Reciprocal of x by Power of x plus Power of a:
 * $\displaystyle \int \frac {\mathrm d x} {x \left({x^n + a^n}\right)} = \frac 1 {n a^n} \ln \left\vert{\frac {x^n} {x^n + a^n} }\right\vert + C$

Setting $n = 2$:

directly.