Q81 / 180 · COMEDK 2024 (Afternoon Shift) · Mathematics

$\int \dfrac{f^{\prime}(x)}{f(x) \log (f(x))} d x \text { is equal to }$

  • Correct answer
    A
    $\log (\log f(x))+C$
  • B
    $\dfrac{1}{\log (\log f(x))}+C$
  • C
    $\dfrac{f(x)}{\log f(x)}+C$
  • D
    $f(x) \log f(x)+C$

Solution

Let $I = \int \dfrac{f'(x)}{f(x) \log(f(x))} dx$.

Substitute $u = \log(f(x))$.

Then, $du = \dfrac{1}{f(x)} \cdot f'(x) dx = \dfrac{f'(x)}{f(x)} dx$.

Substituting these into the integral, we get $I = \int \dfrac{1}{u} du$.

Integrating with respect to $u$, we obtain $I = \log|u| + C$.

Substituting back $u = \log(f(x))$, we get $I = \log|\log(f(x))| + C$.

Answer: $\log (\log f(x))+C$

Hence, the correct answer is option: 'A'.

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