Question

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

Answer 1

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