$\begin{aligned} & \int_0^1 x(1-x)^n d x=\int_0^1(1-x)^n d x\left[\because \int_0^a f(x) d x=\int_0^a f(a-x) d x\right] \\ & =\int_0^1\left(x^n-x^{n+1}\right) d x=\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_0^1 \\ & =\frac{1}{n+1}-\frac{1}{n+2}=\frac{1}{(n+1)(n+2)}\end{aligned}$