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Help for integration

  1. Sep 4, 2012 #1
    1. The problem statement, all variables and given/known data
    [tex]\int_{0}^{∞} x^{2n+1}.e^{-x^2}dx[/tex]
    is equal to (n ε N)
    a)n!
    b)2(n!)
    c)n!/2
    d)(n+1)!/2

    2. Relevant equations



    3. The attempt at a solution
    I can go on solving this by using n=1 or n=2. But i want to do it by a correct method. Is there a proper way to do it? I am having no idea, how should i begin?
     
  2. jcsd
  3. Sep 4, 2012 #2

    uart

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    Science Advisor

    Hi Pranav. Make the substitution [itex]u = x^2[/itex] and see if you can manipulate it in terms of the gamma function.
     
  4. Sep 4, 2012 #3
    Hello uart!

    I have already tried that substitution and i end up with this:
    [tex]\frac{1}{2} \int_{0}^{∞} u^n \cdot e^{-u}du[/tex]
    Gamma function? I guess i have never heard of it.
    Any other way to solve it? If there's no way, i would try to see what this "gamma function" is.
     
  5. Sep 4, 2012 #4

    uart

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    Science Advisor

    Yes, that result is correct. That's good as that makes it equal 1/2 Gamma(n+1). Look up the gamma function and its relation to factorial and you'll find that you basically have the answer there. http://en.wikipedia.org/wiki/Gamma_function
     
  6. Sep 4, 2012 #5

    A_B

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    I used a different method ;integrating x*exp(-ax^2), then differentiation wr to a several times. You'll quickely find the pattern. Then you can prove by induction. I get a result that's not in your options. (but it's late, so I could be wrong)

    A_B
     
  7. Sep 4, 2012 #6
    Thanks a lot uart! :smile:
     
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