Question

Evaluate the integral: \[ \int \frac{\log x}{x} \, dx = ? \]

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

Hint:

When integrating expressions involving \(\log x\) divided by \(x\), substitution \( t = \log x \) simplifies the integral.

Answer 1

The Correct Option is A

Let: \[ I = \int \frac{\log x}{x} \, dx \] Use substitution: \[ t = \log x \implies dt = \frac{1}{x} dx \Rightarrow dx = x \, dt \] Rewrite the integral: \[ I = \int t \, dt = \frac{t^2}{2} + c = \frac{(\log x)^2}{2} + c \]