Invalid interchange of limit and integral

Compute

$$\underset{n\to +\infty }{\lim }{\int }_0^1{n}^{2}x(1-x{)}^{n}dx$$

and deduce that the convergence of the sequence $f_n(x)=n^{2}x(1-x{)}^n$ is not uniform on $[0,1]$.