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Improper Integral Infinity

  1. Mar 12, 2010 #1
    1. The problem statement, all variables and given/known data

    Using the fact that the integral from -Infinity to Infinity of e^-x^2 is equal to Sqrt(Pi), find the integral from -Infinity to Infinity of x^2 * e^-x^2



    2. Relevant equations



    3. The attempt at a solution

    I really dont know how to find this using the fact that the first integral is equal to Sqrt(Pi), where do you start on this one?

    Thanks for the help
     
  2. jcsd
  3. Mar 12, 2010 #2

    phyzguy

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    Phrygian,

    Try integrating by parts with u = x, and dv = x e-x2 dx.
     
  4. Mar 12, 2010 #3
    Thanks a lot! But now how do I evaluate -x/2(e^-x^2) from -Infinity to Infinity?
     
  5. Mar 12, 2010 #4
    The derivative of the argument of the exponential function is, up to a constant factor, in front of the exponential function.
     
  6. Mar 12, 2010 #5
    After doing the integration by parts I ended up with -x/2(e^-x^2) to be evaluated from -infinity to infinity + integral of 1/2 e^-x^2 dx from negative infinity to infinity. I know that the second integral is equal to Sqrt(Pi)/2 but I can't figure out how to evaluate the first part at the limits.
     
  7. Mar 12, 2010 #6

    phyzguy

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    Try writing x e-x2 as x / ex2, then expand the ex2 in the denominator as a power series and watch what happens as x goes to infinity. The ex2 term grows much faster than any power of x.
     
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