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

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- April 15th 2007, 08:21 PMRB06integral proof
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 - April 15th 2007, 08:30 PMJhevon
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)