Question:

$\int x^3 \ln x \, dx$ equals:

(1) $\frac{x^4}{4} \ln x - \frac{x^4}{16} + C$

(2) $\frac{x^4}{4} \ln x + \frac{x^4}{16} + C$

(3) $x^4 \ln x - \frac{x^4}{4} + C$

(4) $\frac{x^4}{4} \ln x - \frac{x^4}{4} + C$

Step-by-Step Solution

Key Concept: Apply integration by parts with $u = \ln x$ and $dv = x^3 \, dx$.

The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.

Correct Answer: (1)