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Math Help - Prove two integral identities???

  1. #1
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    Prove two integral identities???

    1. The following integral identity holds
    \dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{ \sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)x}{a \sqrt{a^{2}-x^{2}}}+x\intop_{x}^{a}\dfrac{d\rho}{\sqrt{\rho^{2  }-x^{2}}}\dfrac{d}{d\rho}\left[\dfrac{F(\rho)}{\rho}\right]
    Hints: this can easily proved by applying ingtegration by parts to the right hand side of the identity
    2. But the following can also hold
    \dfrac{d}{dx}\intop_{x}^{a}\dfrac{F(\rho)d\rho}{ \sqrt{\rho^{2}-x^{2}}}=-\dfrac{F(a)a}{x \sqrt{a^{2}-x^{2}}}+\dfrac{1}{x}\intop_{x}^{a}\dfrac{\rho d\rho}{\sqrt{\rho^{2}-x^{2}}}\dfrac{d}{d\rho}F(\rho)
    I can not figure out the second identity.Is there anybody can help me?I'm waiting for your excellent proof!!
    Last edited by jarvisyang; April 23rd 2013 at 06:00 AM.
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  2. #2
    MHF Contributor
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    Re: Prove two integral identities???

    Hey jarvisyang.

    What substitution did you use for the first integral?
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