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How to prove the value of this integral?

  1. Mar 22, 2013 #1
    [itex]\int^{∞}_{-∞} e^{-x^{2}}dx[/itex] = [itex]\frac{\sqrt{\pi}}{2}[/itex]
    Last edited: Mar 22, 2013
  2. jcsd
  3. Mar 22, 2013 #2


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    Ah, that one. Hint: consider
    ##\int_{-\infty}^{\infty} \int_{-\infty}^\infty e^{-x^2-y^2}\, dx \, dy##
  4. Mar 22, 2013 #3


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    That should be found in pretty much any Calculus text. Look at the integral pwsnafu suggests. Note that, by symmetry, that is [itex]4\int_0^\infty\int_0^\infty e^{-x^2- y^2} dx dy[/itex], over the first quadrant. That can be converted into a "doable" integral by changing to polar coordinates.
  5. Mar 22, 2013 #4
    I don't know Double integrals. Is it possible to prove the result only using Single variable calculus? At first I tried Integration by parts, but I failed :(

    Isn't it possible to integrate using limit of sums and symmetry with suitable manipulations? I tried to sum directly but failed :(
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