Primitive of Function under its Derivative

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

Then:


 * $\ds \int \frac {\map {f'} x} {\map f x} \rd x = \ln \size {\map f x} + C$

where $C$ is an arbitrary constant.

Proof
By Integration by Substitution (with appropriate renaming of variables):
 * $\ds \int \map g u \rd u = \int \map g {\map f x} \map {f'} x \rd x$

Then:

Also presented as
This can also be presented as:
 * $\ds \int \frac {\d u} u = \ln \size u + C$

where it is understood that $u$ is a function of $x$.