Question

Evaluate: ${\int }_0^1{x(1-x)}^{n}dx$

Answer 1

$I={\int }_0^{1}x(1-x{)}^{n}dx$

$={\int }_0^1(1-x)(1-(1-x){)}^{n}dx$ [ Using property of definite integral]

$={\int }_0^1(1-x)(x{)}^{n}dx$

$={\int }_0^1{x}^n-x^{x-1}dx$

$={[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}]}_0^1$

$=\frac{1}{n+1}-\frac{1}{n+2}=\frac{1}{(n+1)(n+2)}$