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Integrating w/o U-Substitution

  1. Apr 30, 2012 #1
    1. The problem statement, all variables and given/known data
    ∫y=e^(-2x^2)dx


    2. Relevant equations



    3. The attempt at a solution
    I can't recall any method for this. I know that the integral of e^x is e^x, but I know that in this case the integral would not be the same as the original function because the derivative of -2x^2 would be -4x. Can anyone help me?
     
  2. jcsd
  3. Apr 30, 2012 #2

    Curious3141

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    Homework Helper

    Short answer: You can't evaluate that integral in the usual way.

    Long answer: This is related to the Gaussian integral:

    [tex]\int_{-\infty}^{\infty}e^{-x^2}dx[/tex]

    which you should be able to see by making the substitution [itex]u = \sqrt{2}x[/itex] into your integral.

    The indefinite integral of the function [itex]e^{-x^2}[/itex] cannot be expressed in terms of elementary functions. There is a related function, called the error function, written as [itex]erf(x)[/itex], which involves a definite integral of the same function between finite bounds [itex]0[/itex] and [itex]x[/itex]. Again, the error function cannot be expressed in terms of elementary functions, and is simply defined in terms of that integral.
     
  4. Apr 30, 2012 #3
    definite integral of it can be evaluated between limits 0 to infinity by transforming to polar coordinates.
     
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