Primitive of x over x squared plus a squared

Theorem

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

where $a$ is a non-zero constant.

Proof
Let:

As $x^2 + a^2 > 0$ for all $x \in \R$ there is no need to invoke the absolute value of the operand of the $\ln$ function.