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Homework Help: Improper integral x^(2) * e^(-x^2)

  1. Aug 26, 2010 #1
    1. The problem statement, all variables and given/known data

    Show that [tex]\int_{0}^{\infty}x^{2}e^{-x^{2}}dx = \frac{1}{2}\int_{0}^{\infty}e^{-x^{2}}dx.[/tex]

    2. Relevant equations

    None.

    3. The attempt at a solution

    I used substitution:

    [tex]t = x^{2}[/tex]

    [tex]dx = \frac{dt}{2x}[/tex]

    [tex]\frac{1}{2}\int_{0}^{\infty}\sqrt{t}e^{-t}dx[/tex]

    Then tried using integration by parts but then I didnt get an answer and got stuck.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Aug 26, 2010 #2

    lanedance

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    Homework Helper

    have you tried straight integartion by parts?
     
  4. Aug 26, 2010 #3
    Try:

    x^2*e^(-x^2) = [-0.5*x]*[-2*x*e^(-x^2)dx]

    [-0.5*x] = u
    [-2*x*e^(-x^2)dx] = dv

    Tell us if you got it !

    Good Luck :)
     
  5. Aug 26, 2010 #4
    I tried integration by parts but it didnt work out. I only learn how to integrate e^(-x^2) next year.
     
  6. Aug 26, 2010 #5

    HallsofIvy

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    Science Advisor

    How did you try? Did you try letting [itex]u= x[/itex] and [itex]dv= xe^{-x^2}dx[/itex].
     
  7. Aug 26, 2010 #6
    Try integration by parts with t=x, not t=x^2 !

    I remind you:

    integral(u*dv) = u*v - integral(v*du)

    P.S.: If you learn in the next year how to get the anti-derivative of e^(-x^2), please tell us ;)
     
    Last edited: Aug 26, 2010
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