Question

Solve:${\int }_0^{1}x(1-x{)}^{n}dx$

Answer 1

Let $1-x=t$, then $dx=-dt$
When $x=0,t=1;x=1,t=0$
${\int }_1^0(1-t)t^n(-dt)$
$={\int }_0^1(t^n-t^{n+1})dt$
$=\frac{1}{n+1}-\frac{1}{n+2}$
$=\frac{1}{(n+1)(n+2)}$