For \(x \gt e\), \(\int\displaystyle\frac{dx}{x\log x\,\log\left(\log x\right)}\) is equal to
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\log\left|\log\left(\log x\right)\right|+C\)
Let \(t = \log\left(\log x\right)\). By the chain rule, its derivative is exactly the rest of the integrand:
\( \displaystyle dt = \frac{1}{\log x}\cdot\frac{1}{x}\,dx = \frac{dx}{x\log x} \)
So the integral becomes
\( \displaystyle \int\frac{dt}{t} = \log\left|t\right|+C \)
\( = \log\left|\log\left(\log x\right)\right|+C \)