Question

Evaluate the integral \( \displaystyle \int \frac{1}{x\log x}\,dx \).

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

Hint:

A very common integral result: \[ \int \frac{f'(x)}{f(x)}dx = \log|f(x)| + C \] Recognizing this pattern makes many logarithmic integrals easy to solve.

Answer 1

The Correct Option is B

Concept: When an integral contains a function and its derivative, substitution is a useful method. If \(u = f(x)\), then: \[ \int \frac{f'(x)}{f(x)}dx = \log|f(x)| + C \]

Step 1: Use substitution. Let \[ u = \log x \] Then \[ \frac{du}{dx} = \frac{1}{x} \] \[ du = \frac{1}{x}dx \]

Step 2: Substitute into the integral. \[ \int \frac{1}{x\log x}dx \] \[ = \int \frac{1}{u}du \]

Step 3: Integrate. \[ \int \frac{1}{u}du = \log|u| + C \]

Step 4: Substitute back \(u = \log x\). \[ \log|u| + C = \log|\log x| + C \] Hence, \[ \boxed{\log|\log x| + C} \]