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Thread: Help for proof an integral

  1. #1
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    Help for proof an integral

    Hello,
    Can anybody help me to proof the following integrals?

    Help for proof an integral-capture.png
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  2. #2
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    Re: Help for proof an integral

    Hey ariabob.

    Hint - look at the fundamental theorem of calculus.
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  3. #3
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    Re: Help for proof an integral

    Hi chiro,
    I still could not reach to the proof.
    Iam stuck with the proof I equations
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  4. #4
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    Re: Help for proof an integral

    For the first one:
    $u = \dfrac{1}{1+Gr^{-\alpha}}, dv = 2r\, dr \Longrightarrow du = \dfrac{Gr^{-\alpha-1}}{\left(1+Gr^{-\alpha}\right)^2 } dr, v = r^2$

    $\begin{align*}\displaystyle \dfrac{1}{R^2}\int_0^R \dfrac{2r\, dr}{1+Gr^{-\alpha}} & = \dfrac{1}{R^2}\left( \dfrac{R^2}{1+GR^{-\alpha}} - \int_0^R \dfrac{Gr^{1-\alpha}\, dr}{\left(1+Gr^{-\alpha}\right)^2}\right) \\ & = \dfrac{1}{1+GR^{-\alpha}} - \dfrac{G}{R^2} \int_0^R \dfrac{r^{1-\alpha}\, dr}{\left(1+Gr^{-\alpha}\right)^2}\end{align*}$

    Keep using integration by parts to yield an infinite sequence. At this point, hopefully, it will resemble the hypergeometric function you are looking for.
    Last edited by SlipEternal; Jan 9th 2018 at 06:47 AM.
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