Question

\( \int \frac{(\log x)^2}{x} \, dx \) equals:

• A. \( \frac{1}{x} + C \)
• B. \( \log x + C \)
• C. \( \frac{(\log x)^2}{2} + C \)
• D. \( \frac{(\log x)^3}{3} + C \)

Hint:

For integrals involving logarithms, use substitution \( u = \log x \) to simplify the integral.

Answer 1

The Correct Option is D

Step 1: Simplifying the integral. 
We are given the integral: \[ I = \int \frac{(\log x)^2}{x} \, dx \] This is a standard integral that can be solved by substitution. 
Step 2: Substitution. 
Let \( u = \log x \), so that \( du = \frac{1}{x} dx \). The integral becomes: \[ I = \int u^2 \, du \] Step 3: Integrating. 
Now, integrate with respect to \( u \): \[ I = \frac{u^3}{3} + C = \frac{(\log x)^3}{3} + C \] Step 4: Conclusion. 
Thus, the integral is \( \frac{(\log x)^3}{3} + C \), corresponding to option (d).