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Integrating absolute values over infinity

  1. Nov 7, 2012 #1

    ElijahRockers

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    1. The problem statement, all variables and given/known data

    Find <x> in terms of X0 if X0 is constant and

    [itex]\Psi(x) = \frac{1}{\sqrt{X_0}}e^{\frac{-|x|}{X_0}}[/itex]

    and

    [itex]<x> = \int^{\infty}_{-\infty}{\Psi^* x \Psi}dx[/itex]

    where Psi* is the complex conjugate of Psi.

    Since there is no imaginary component, this is effectively Psi2.

    so, from here I could do a u-substitution to integrate over e^u du, but I'm not sure how.

    What is the derivative of -2|x|/X_0 with respect to x?

    This is part of a physics exercise i'm working on.

    [itex] <x> = \frac{1}{x_0}\int^{\infty}_{-\infty}e^{\frac{-2|x|}{X_0}} dx[/itex]

    I have found that the derivative of |x| depends on whether x<0 or x>0. for x<0, x'=-1 and for x>0, x'=1 but I'm not sure how to tie this all together for the integration.

    I guess what I'm really asking is how do I find the integrand here?
     
    Last edited: Nov 7, 2012
  2. jcsd
  3. Nov 7, 2012 #2

    SammyS

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    So, split the integral.

    [itex]\displaystyle <x> = \frac{1}{x_0}\int^{0}_{-\infty}xe^{\frac{-2|x|}{X_0}} dx+\frac{1}{x_0}\int^{\infty}_{0}xe^{\frac{-2|x|}{X_0}} dx[/itex]


    Added in Edit:

    And where did the x go in the integrand ? Well, that makes the integrand odd. Therefore, don't bother splitting it.
     
    Last edited: Nov 7, 2012
  4. Nov 7, 2012 #3

    Dick

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    Just integrate from 0 to infinity where |x|=x. The integral from -infinity to 0 will be the same thing since your function is even.
     
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