Is $ \int \frac{f'(x)}{f(x)} \ dx = \log |f(x)| + C$ true for all differentiable functions $f$? Let $f$ be a differentiable function. Is the following identity true for all such $f$?
$$
\int \frac{f'(x)}{f(x)} \ dx = \log |f(x)| +  C
$$
I ask because there exist differentiable functions whose derivatives are not Riemann integrable (see here for instance). On the other hand, if we use the substitution $u = f(x)$ for $f$ on $[a,b]$,
$$
\int_a^b \frac{f'(x)}{f(x)} \ dx  = \int_{f(a)}^{f(b)} \frac{1}{u} \ du
$$
and the RHS appears to be integrable. How can we reconcile this?
Any comments, help and explanations are welcome.

Be careful too. If $f(x) = 0$ for any $x \in [a,b]$, the integral won't be defined.
