The value of the integral I = _0^1 x(1-x)^n,dx is:

Question

The value of the integral $I = \displaystyle\int_0^1 x(1-x)^n\,dx$ is:

Accepted Answer

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

$$ = \int\limits_0^1 {\left( {1 - x} \right){{\left( {1 - 1 + x} \right)}^n}dx} $$

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

$$ = \left[ {{{{x^{n + 1}}} \over {n + 1}} - {{{x^{n + 2}}} \over {n + 2}}} \right]_0^1$$

$$ = {1 \over {n + 1}} - {1 \over {n + 2}}$$

Solutions · 1