Question

Find the value of `int_0^1 x(1 - x)^n dx`.

Answer 1

Suppose

I = `int_0^1 x(1 - x)^n dx`

⇒ I = `int_0^1 (1 - x)[1 - (1 - x)]^n dx`   ...[By King Rule]

⇒ I = `int_0^1 (1 - x)x^n dx`

⇒ I = `int_0^1 (x^n - x^(n + 1)) dx`

= `[x^(n + 1)/(n + 1) - x^(n + 2)/(n + 2)]_0^1`

⇒ I = `[1/(n + 1) - 1/(n + 2)] - [0 - 0]`

= `1/((n + 1)(n + 2))`