$\int\limits_0^1 x {\left( {1 - x} \right)^n}dx$

Let $I = \int\limits_0^1 x {\left( {1 - x} \right)^n}dx$

$\Rightarrow$ $I = \int\limits_0^1 {\left( {1 - x} \right){{\left[ {1 - \left( {1 - x} \right)} \right]}^n}dx}$

$= \int\limits_0^1 {\left( {1 - x} \right){x^n}dx}$

$= \int\limits_0^1 {\left( {{x^n} - {x^{n + 1}}} \right)dx} = \left[ {\cfrac{{{x^{n + 1}}}}{{n + 1}} - \cfrac{{{x^{n + 2}}}}{{n + 2}}} \right]_0^1$

$= \left( {\cfrac{1}{{n + 1}} - \cfrac{1}{{n + 2}}} \right) - \left( {0 - 0} \right) = \cfrac{1}{{\left( {n + 1} \right)\left( {n + 2} \right)}}$