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Math Help - improper integration by parts

  1. #1
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    improper integration by parts

    I need to prove the following:
    \int^\infty_0 x^2e^{-x^2}=\frac{1}{2}\int^\infty_{0} e^{-x^2}
    Im having trouble integrating this, although I do know, based on the Gamma function, that \frac{1}{2}\int^\infty_{0} e^{-x^2}=\frac{\sqrt{\pi}}{4}
    So really what I need to show is that \lim_{b\to \infty}\int^b_0 x^2e^{-x^2}=\frac{\sqrt{\pi}}{4}
    But when I try to integrate by parts, Im getting even more confusing terms, can someone help me out?
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  2. #2
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    Question

    When you're doing integration by parts, what are you letting u equal and v equal in the formula?
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  3. #3
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    u = x^2, dv= \int^\infty_0 e^{-x^2}dx
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  4. #4
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    Try integration by parts with

    u = x
    dv = x e^{-x^2} \, dx.
    Last edited by awkward; December 8th 2009 at 04:47 PM. Reason: left out a d
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  5. #5
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    so... I did the integration by parts according to these suggestions, and what I came up with, supposing I did the integration correctly, is:
    \frac{1}{4}\lim_{b\to\infty}[b-3be^{-b^2}]
    I need this to equal \frac{\sqrt{\pi}}{4}...
    So either I am missing something or the integration didnt go as it should have.
    Any feedback? thanks
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  6. #6
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    I don't know why you need to worry about the actual value- if you let u= x, dv= xe^{-x^2}, integration by parts puts \int_0^\infty e^{-x^2}dx in your lap! I can only conclude that you have done the integration by parts incorrectly. Please show all your work.
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