Question

Evaluate \[ \int x^2(\log x)^2\,dx. \]

• A. \[ \frac16x^3\left[3(\log x)^2-3\log x+4\right]+C \]
• B. \[ \frac1{27}x^3\left[9(\log x)^2-6\log x+2\right]+C \]
• C. \[ \frac1{27}x^3\left[9(\log x)^2+6\log x+2\right]+C \]
• D. \[ \frac19x^3\left[6(\log x)^2-3\log x+1\right]+C \]

Hint:

For integrals involving powers of \(\log x\), choose \((\log x)^n\) as the first function in integration by parts. Each application reduces the power of \(\log x\) by one.

Answer 1

The Correct Option is B