Primitive of x over a x + b/Proof 2

Theorem

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

Proof
From Reduction Formula for Primitive of Power of $x$ by Power of $a x + b$: Decrement of Power of $x$:
 * $\displaystyle \int x^m \left({a x + b}\right)^n \ \mathrm d x = \frac {x^m \left({a x + b}\right)^{n + 1} } {\left({m + n + 1}\right) a} - \frac {m b} {\left({m + n + 1}\right) a} \int x^{m - 1} \left({a x + b}\right)^n \ \mathrm d x$

Let $m = 1$ and $n = -1$.

Then: