Question:
Evaluate $\int x^{3}(\log x)^{2} d x$
Solution:
Use method of integration by parts
$y=\log ^{2} x \int x^{3} d x-\int \frac{d}{d x} \log ^{2} x\left(\int x^{3} d x\right) d x$
$y=\log ^{2} x \frac{x^{4}}{4}-\int \frac{2 \log x}{x} \frac{x^{4}}{4} d x$
$y=\frac{x^{4}}{4} \log ^{2} x-\frac{1}{2}\left(\log x \int x^{3} d x-\int \frac{d}{d x} \log x\left(\int x^{3} d x\right) d x\right.$
$y=\frac{x^{4}}{4} \log ^{2} x-\frac{1}{2}\left(\log x \frac{x^{4}}{4}-\int \frac{1}{x} \frac{x^{4}}{4} d x\right)$
$y=\frac{x^{4}}{4} \log ^{2} x-\frac{x^{4}}{8} \log x+\frac{x^{4}}{32}+c$