$\int \frac{f'(x)}{f(x) \log_e[f(x)]} \, dx$ is equal to

\( \log_e(\log_e[f(x)]) + C \); \( C \) is a constant of integration

The correct answer is Option (3) → \( \log_e(\log_e[f(x)]) + C \); \( C \) is a constant of integration

Given integral:

$\int \frac{f'(x)}{f(x) \log_e[f(x)]}\,dx$

Substitute: $u = \log_e[f(x)]$

$\Rightarrow \frac{du}{dx} = \frac{f'(x)}{f(x)}$

$\Rightarrow du = \frac{f'(x)}{f(x)}\,dx$

So the integral becomes:

$\int \frac{1}{u}\,du = \log_e|u| + C = \log_e|\log_e[f(x)]| + C$