$\int\limits _0^1 x(1-x)^n dx$
=>$ 1-x=t=>-dx=dt$
at $x=0$. upper limit=1
at $x=1$. lower limit=0
=> $\int \limits _1^0 (1-t).f^n.(-dt)$
=> $- \int \limits_0^1 t^n.dt+ \int \limits _1^0 t^{n+1}.dt$
=> $ -\bigg(\large\frac{t^{n+1}}{n+1}\bigg)^0_1+\frac{t^{n+2}}{n+2}\bigg]_1^0$
=> $ \bigg(\large\frac{1}{n+1}$$+0-\large\frac{1}{n+2} \bigg)$