Hello, can someone please help me on this problem? Any suggestions is welcomed
Given that n is a positive integer, evaluate the integral
int_0^{1} x(1-x)^n dx
Thanks
int{x(1 - x)^n}dx
we proceed by substitution
let u = 1 - x
=> du = -dx
=> -du = dx
now if u = 1 - x => x = 1 - u
so our integral becomes:
-int{(1 - u)u^n}du = -int{u^n - u^(n+1)}du
...........................= [{u^(n+1)}/(n+1) - {u^(n+2)}/(n+2)]
...........................= [{(1-x)^(n+1)}/(n+1) - {(1-x)^(n+2)}/(n+2)] evaluated between 0 and 1
...........................= -(1/(n+1) - 1/(n+2))
...........................= 1/(n+2) - 1/(n+1)
...........................= (n + 1 - n - 2)/(n+1)(n+2)
...........................= -1/(n+1)(n+2)