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Math Help - Annoying integral

  1. #1
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    Annoying integral

    Let  f(x) = \log{(1-x^2})

    Let  I_n = \int_{n-\frac{1}{2}}^{n+\frac{1}{2}} \log{x} \ dx - \log{n} for n natural.

    Show that  I_n = \int_0^{\frac{1}{2}} f\left(\frac{t}{n}\right) dt

    I just can't get this. Any help would be appreciated. Thanks
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  2. #2
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    Jan 2009
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    Quote Originally Posted by Mathmo55 View Post
    Let  f(x) = \log{(1-x^2})

    Let  I_n = \int_{n-\frac{1}{2}}^{n+\frac{1}{2}} \log{x} \ dx - \log{n} for n natural.

    Show that  I_n = \int_0^{\frac{1}{2}} f\left(\frac{t}{n}\right) dt

    I just can't get this. Any help would be appreciated. Thanks

      \int_{n-\frac{1}{2}}^{n+\frac{1}{2}} =  \int_{n-\frac{1}{2}}^n + \int_n^{n+\frac{1}{2}}


    For the first one , substitute  x = n-t

    For the second one , substitute  x = n+t


    I think we will see such thing :

     I_n = \int_0^{\frac{1}{2}} [\log(n-t) + \log(n+t)]~dt  -\log{n}

     = \int_0^{\frac{1}{2}} \log(n^2-t^2) ~dt  -\log{n}

     = \int_0^{\frac{1}{2}}  [\log(n^2) + f(\frac{t}{n}) ]~dt -\log{n}

     = \log{n} + \int_0^{\frac{1}{2}} f(\frac{t}{n})~dt - \log{n}

     = \int_0^{\frac{1}{2}} f(\frac{t}{n})~dt
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