Question

Evaluate$\underset{0}{\overset{1}{\int }}x{\left(1-x\right)}^{n}dx$

Answer 1

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

Let $t=1-x\Rightarrow dt=-dx$

When $x=0\Rightarrow t=1$

When $x=1\Rightarrow t=0$

$=-{\int }_1^0(1-t)t^{n}dt$

$=-{\int }_1^0(t^n-t^{n+1})dt$

$=-{[\frac{t^{n+1}}{n+1}-\frac{t^{n+2}}{n+2}]}_1^0$

$=-[\frac{1^{n+1}}{n+1}-\frac{1^{n+2}}{n+2}-0]$

$=-[\frac{1}{n+1}-\frac{1}{n+2}]$

$=-\frac{n+2-n-1}{(n+1)(n+2)}$

$=\frac{-1}{(n+1)(n+2)}$