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Math Help - integral proof

  1. #1
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    integral 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
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by RB06 View Post
    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)
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