\(\int\displaystyle\frac{f'(x)}{f(x)\log_e[f(x)]}\, dx\) is equal to
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\log_e(\log_e[f(x)]) + C\): \(C\) is a constant of integration
The integrand contains \(\frac{f'(x)}{f(x)}\), the derivative of \(\log_e[f(x)]\), so a substitution works. Let
\(t = \log_e[f(x)]\)
\( \displaystyle dt = \frac{f'(x)}{f(x)}\,dx \)
\( \displaystyle \int \frac{f'(x)}{f(x)\log_e[f(x)]}\,dx = \int \frac{dt}{t} = \log_e|t| + C \)
\(= \log_e(\log_e[f(x)]) + C\)
(The options write it without the modulus sign, which is the form for \(\log_e[f(x)] \gt 0\).)
Shortcut: Differentiate Option (3) by the chain rule: \(\displaystyle\frac{1}{\log_e[f(x)]} \cdot \frac{f'(x)}{f(x)}\), which is exactly the integrand.