Improper Integral Infinity

  • Thread starter phrygian
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  • #1
phrygian
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Homework Statement



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



Homework Equations





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
 

Answers and Replies

  • #2
phyzguy
Science Advisor
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Phrygian,

Try integrating by parts with u = x, and dv = x e-x2 dx.
 
  • #3
phrygian
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Thanks a lot! But now how do I evaluate -x/2(e^-x^2) from -Infinity to Infinity?
 
  • #4
Count Iblis
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The derivative of the argument of the exponential function is, up to a constant factor, in front of the exponential function.
 
  • #5
phrygian
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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.
 
  • #6
phyzguy
Science Advisor
5,026
2,025
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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