Question

By using properties of definite integrals, evaluate the integrals
${\int }_0^{1}x(1-x{)}^{n}dx.$

Answer 1

Let ${\int }_0^{1}x(1-x{)}^{n}dx.$
$\Rightarrow I={\int }_0^1(1-x){\left\{\right(1-x)1-(1-x\left)\right\}}^{n}dx[\because {\int }_0^{a}f(x)dx={\int }_0^{a}f(a-x\left)dx\right]$
$={\int }_0^1(1-x)x^{n}dx=={\int }_0^1(x^n-x^{n+1})dx$
$={[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}]}_0^1=[\frac{1}{n+1}-\frac{1}{n+2}]=0$
$=\frac{(n+2)-(n+1)}{(n+1)(n+2)}=\frac{1}{(n+1)(n+2)}$