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Thread: Integral problem

  1. #1
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    Integral problem

    Need help with some aspects of this problem (particularly the last part). Would greatly appreciate any help offered.

    'For $n \geq 0$, let

    $I_n=\int\limits_0^1{x}^{n}(1-x)^n\, \mathrm{d}x $

    For $n \geq 1$, show, by means of a substitution, that

    $\int\limits_0^1{x}^{n-1}(1-x)^n\, \mathrm{d}x=\int\limits_0^1{x}^n(1-x)^{n-1}\, \mathrm{d}x $

    and deduce that

    $2\int\limits_0^1{x}^{n-1}(1-x)^n\, \mathrm{d}x=I_{n-1}$

    Show also, for $n \geq 1$, that

    $I_n=\frac{n}{n+1}\int\limits_0^1{x}^{n-1}(1-x)^{n+1}\, \mathrm{d}x $

    and hence that

    $I_n=\frac{n}{2(2n+1)}I_{n-1}$.'




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  2. #2
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    Re: Integral problem

    A fairy obvious substitution is u= 1- x. Of course the dummy variable of integration on the right would then be "u" which you could change to "x".
    Thanks from topsquark
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  3. #3
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    Re: Integral problem

    Yes, thanks Halls, no problem with the first.
    Would this be correct for the next?:

    $2\int\limits_0^1{x}^{n-1}(1-x)^{n}\, \mathrm{d}x = \int\limits_0^1{x}^{n-1}(1-x)^n\, \mathrm{d}x + \int\limits_0^1{x}^n(1-x)^{n-1}\, \mathrm{d}x = \int\limits_0^1{x}^{n-1}(1-x)^{n} + {x}^{n}(1-x)^{n-1}\, \mathrm{d}x = \int\limits_0^1{x}^{n-1}(1-x)^{n-1}(1-x+x) = I_{n-1}$
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  4. #4
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    Re: Integral problem

    Ok, I'm alright with this problem apart from the last part. Can anybody help?
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  5. #5
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    Re: Integral problem

    Quote Originally Posted by jimi View Post
    Ok, I'm alright with this problem apart from the last part. Can anybody help?
    Integration by parts

    $\displaystyle u=x^n $

    $\displaystyle dv=(1-x)^ndx$
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