Primitive of Arctangent of x over a/Proof 2

Theorem

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

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

and let:

Then: