Question

Evaluate:

`int_0^1x(1 - x)^ndx`; (where n ∈ N).

Answer 1

`int_0^1x(1 - x)^ndx`

= `int_0^1(1 - x){1 - (1 - x)}^ndx,  ...(As, int_0^af(x)dx = int_0^af(a - x)dx)`

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

= `int_0^1x^ndx - int_0^1x^(n + 1)dx`

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

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

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