The integral ∫limits_0^1 x(1-x)^n d x is equal to :

The correct answer is Option (2) - $\frac{1}{(n+1)(n+2)}$

$I=\int\limits_0^1 x(1-x)^{n} dx$

$=\int\limits_0^1 (1-x)(x)^ndx$

$=\int\limits_0^1x^n-x^{n+1}dx=\left[\frac{x^{n+1}}{n+1}-\frac{x^{n+2}}{n+2}\right]_0^1$

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