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Help to prove a reduction formula

  1. Sep 1, 2007 #1

    rock.freak667

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    Homework Helper

    1. The problem statement, all variables and given/known data
    Let [tex]I_n=\int_{0}^{1} (1+x^2)^{-n} dx[/tex] where [tex]n\geq1[/tex]
    Prove that [tex]2nI_{n+1}=(2n-1)I_n+2^{-n}


    2. Relevant equations

    consider:
    [tex] \frac{d}{dx}(x(1+x^2)^n) [/tex]


    3. The attempt at a solution

    [tex]\frac{d}{dx}(x(1+x^2)^n = (1+x^2)-2nx^2(1+x^2)^{-n-1}[/tex]

    Integrating both sides between 1 and 0

    [tex] \left[ x(1+x^2)^n \right]_{0}^{1} = I_n -2n\int_{0}^{1} x^2(1+x^2)^{-n-1}[/tex]

    [tex]2n\int_{0}^{1} x^2(1+x^2)^{-n-1}[/tex] = [tex]\left[ \frac{x^2(1+x^2)^{-n-1}}{2} \right]_{0}^{1} + (n+1)\int_{0}^{1} x^3(1+x^2)^{-n-2}[/tex]

    which is even more of story to integrate by parts..is there any easier way to integrate[tex]2nx^2(1+x^2)^{-n-1}[/tex] ?
     
    Last edited: Sep 1, 2007
  2. jcsd
  3. Sep 2, 2007 #2

    Avodyne

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    Science Advisor

    Try expressing the integral of [tex]2nx^2(1+x^2)^{-n-1}[/tex] in terms of [tex]I_{n+1}[/tex] and [tex]I_{n}[/tex].
     
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