Evaluate the following integrals:

Let $I=\int x^{3} \log x d x$

Using integration by parts,

$I=\log x \int x^{3} d x-\int \frac{d}{d x} \log x \int x^{3} d x$

We have, $\int x^{n} d x=\frac{x^{n+1}}{n+1}$ and $\int \log x=\frac{1}{x}$

$=\log x \times \frac{x^{4}}{4}-\int \frac{1}{x} \times \frac{x^{4}}{4}$

$=\log x \times \frac{x^{4}}{4}-\frac{1}{4} \int x^{3} d x$

$=\frac{x^{4}}{4} \log x-\frac{1}{4} \times \frac{x^{4}}{4}$

$=\frac{x^{4}}{4} \log x-\frac{x^{4}}{16}+c$