Evaluate the following integrals:

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

Using integration by parts,

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

We know that,

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

$=\log x \frac{x^{n+1}}{n+1}-\int \frac{1}{x} \times \frac{x^{n+1}}{n+1} d x$

$=\log x \frac{x^{n+1}}{n+1}-\int \frac{x^{n}}{n+1} d x$

$=\log x \frac{x^{n+1}}{n+1}-\frac{1}{n+1}\left[\int x^{n} d x\right]$

We know that,

$\int x^{n} d x=\frac{x^{n+1}}{n+1}$

$=\log x \frac{x^{n+1}}{n+1}-\frac{1}{(n+1)^{2}} x^{n+1}+c$