Question

Find the value of ${\int }_0^{1}x(1-x{)}^{n}dx.$

Answer 1

$\begin{array}{c}I={\int }_0^{1}x{\left(1-x\right)}^{n}dx\\ t=1-x\Rightarrow dt=-dx\\ x\to 0t\to 1\\ x\to 1t\to 0\\ \Rightarrow -{\int }_1^0\left(1-t\right)t^{n}dt\\ \Rightarrow {\int }_0^1\left(t^n-t^{n+1}\right)dt\Rightarrow {\left[\frac{t^{n+1}}{n+1}-\frac{t^{n+2}}{n+2}\right]}_0^1\\ \Rightarrow \frac{1}{n+1}-\frac{1}{n+2}=\frac{1}{\left(n+1\right)\left(n+2\right)}\end{array}$