Question

\(\int \frac{dx}{x \log x} \)

• A. \( \log x + k \)
• B. \( (\log x)^2 + k \)
• C. \( \log(\log x) + k \)
• D. \( \frac{1}{\log x} + k \)

Hint:

Use substitution for integrals involving logarithmic terms. Here, substituting \( u = \log x \) simplifies the problem.

Answer 1

The Correct Option is C

We are given the integral: \[ I = \int \frac{dx}{x \log x} \] This is a standard integral, which can be solved using the substitution method. Let: \[ u = \log x \quad \text{so that} \quad du = \frac{dx}{x} \] Substituting this into the integral: \[ I = \int \frac{du}{u} = \log u + k = \log (\log x) + k \] Thus, the correct answer is \( \log(\log x) + k \).