Question

Evaluate the integral \( \int x^3 (\log x)^2 dx \):

• A. \( \frac{(\log x)^2 x^4}{4} + \frac{1}{2} \left[ (\log x) x^4 - \frac{x^4}{16} \right] + C \)
• B. \( \frac{(\log x)^2 x^4}{4} - \frac{1}{2} \left[ (\log x) x^4 - \frac{x^4}{16} \right] + C \)
• C. \( \frac{(\log x)^2 x^4}{4} + \frac{1}{2} \left[ (\log x) x^4 - \frac{x^4}{4} \right] + C \)
• D. \( \frac{(\log x)^2 x^4}{4} + \frac{1}{2} \left[ (\log x) x^4 - \frac{x^4}{16} \right] + C \)

Hint:

For integrals involving logarithmic functions, use integration by parts and keep track of constants carefully.

Answer 1

The Correct Option is A

To solve the integral \( \int x^3 (\log x)^2 dx \), we use integration by parts. Let: \[ u = (\log x)^2 \quad \text{and} \quad dv = x^3 dx \] Then, \( du = \frac{2 \log x}{x} dx \) and \( v = \frac{x^4}{4} \). Now apply the integration by parts formula: \[ \int u dv = uv - \int v du \] Substitute the values of \( u \), \( du \), \( v \), and \( dv \), and integrate the resulting expression. After integrating, we obtain: \[ \frac{(\log x)^2 x^4}{4} + \frac{1}{2} \left[ (\log x) x^4 - \frac{x^4}{16} \right] + C \] Thus, the correct answer is option (1).