Primitive of Function under its Derivative

Theorem
Let $f$ be a real function which is integrable.

Then:


 * $\displaystyle \int \frac {f' \left({x}\right)} {f \left({x}\right)} \ \mathrm d x = \ln \left|{f \left({x}\right)}\right| + C$

where $C$ is an arbitrary constant.

Proof
By Integration by Substitution (with appropriate renaming of variables):
 * $\displaystyle \int g \left({u}\right) \ \mathrm d u = \int g \left({f \left({x}\right)}\right) f' \left({x}\right) \ \mathrm d x$

Then: