I=\int _0^1\frac{x^n}{n}\ln  xdx=\left|\begin{array}{c} u=\text{lnx} \\ \text{dv}=\frac{x^n}{n}\end{array}\begin{array}{c}
 \text{du}=\frac{\text{dx}}{x} \\
 v=\frac{x^{n+1}}{n(n+1)}\end{array}\right|=\frac{x^{n+1}}{n(n+1)}\text{lnx}|_0^1-\int_0^1 \frac{x^n}{n(n+1)} \, dx=-\lim_{x\to 0} \frac{x^{n+1}}{n(n+1)}\text{lnx}-\frac{x^{n+1}}{n(n+1)^2}|_0^1=\frac{-1}{n(n+1)^2}