I need to prove that $$\int \frac{f'(x)}{f(x)} = \ln\left(|f(x)|\right) + C.$$ Working in $\mathbb{R}$.
My development was:
Deriving the RHS: $$ { \frac{d}{dx}\ln\left(|f(x)|\right) = \frac{1}{|f(x)|} \cdot \frac{f(x)}{|f(x)|} \cdot f'(x) = \frac{f(x)f'(x)}{f^2(x)} =\frac{f'(x)}{f(x)} } $$ and therefore $${ \ln\left(|f(x)|\right)} \ \ \ \text{ is the primitive of } \ \ {\frac{f'(x)}{f(x)}}. $$
First, i need to check if my solution is correct. On the other hand, i want to know if there are other reasons(different of the common argument that says that logarithm take positive values) to take $\ln(|f(x)|) + C$ as the primitive instead of $\ln(f(x)) + C.$
Thanks in advance.