Evaluate: ∫_0^1 x(1-x)^n d x

\[\begin{aligned} I & =\int_{0}^{1} x(1-x)^{n} d x \\ & =\int_{0}^{1}(1-x)[1-(1-x)]^{n} d x \quad \text { [using property] } \\ & =\int_{0}^{1} x^{n}(1-x) d x \\ = & \int_{0}^{1} x^{n} d x-\int_{0}^{1} x^{n+1} d x \\ = & {\left[\frac{x^{n+1}}{n+1}\right]_{0}^{1}-\left[\frac{x^{n+2}}{n+2}\right]_{0}^{1} } \\ & =\frac{1}{n+1}-\frac{1}{n+2} \text { Or } \frac{1}{(n+1)(n+2)} \end{aligned} \]