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Need hints on how to integrate this

  1. Apr 3, 2008 #1
    Hello,

    I need to evaluate the following integral with respect to x. If you guys can give me some hints on how to start off, it would be greatly appreciated.

    [tex]f{x}[/tex](x) = [tex]\int\frac{\alpha\beta}{4}e^{(-\alpha|x| - \beta|y|)}dy[/tex]

    The integral is evaluate from -infinity to infinity

    From this I know that [tex]\frac{\alpha\beta}{4}[/tex] is a constant and can be taken out. I get confused on how to integrate the absolute value portion of it.

    Thanks!
     
  2. jcsd
  3. Apr 3, 2008 #2
    I did some manipulation and am at the following part:

    [tex]\frac{\alpha\beta}{4}[/tex][tex]\int e^{-\alpha|x|} e^{-\beta|y|} dy[/tex]

    =[tex]\frac{\alpha\beta}{4}e^{-\alpha|x|}[/tex][tex]\int e^{-\beta|y|} dy[/tex]
    evaluate from -infinity to infinity
     
    Last edited: Apr 3, 2008
  4. Apr 3, 2008 #3

    Gib Z

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    Homework Helper

    To where you are now, evaluating the integral should be easy, noting that the remaining integrand is an even function.
     
  5. Apr 4, 2008 #4
    Figured it out. Thanks!
     
  6. Apr 4, 2008 #5
    =[tex]\frac{\alpha\beta}{4}e^{-\alpha|x|}[/tex][tex]\int e^{-\beta|y|} dy[/tex]
    evaluate from -infinity to infinity

    Since this is an even function, we can just multiply the function by 2 and take out the absolute value because of symmetry and taking the limits of integration accordingly.

    Therefore,
    [tex]\frac{\alpha\beta}{4}e^{-\alpha|x|} * 2[/tex] [tex] \int e^{-\beta y} dy [/tex]
    I have attached what this graph would look like as a function of x. There is symmetry about the origin and the graph is even, so this function is even. We can just multiply the function by 2 and evaluate the integral from 0 to infinity because of the symmetry.

    =[tex]\frac{\alpha\beta}{2}e^{-\alpha|x|}[/tex] [tex] \int e^{-\beta y} dy [/tex], evalualted from 0 to infinity.

    This integral evaluates to:
    [tex]\frac{\alpha\beta}{2}e^{-\alpha|x|}[/tex] * [tex]\left(\frac{-1}{\beta}e^{-\beta y}\right)[/tex] evaluated from 0 to infinity.

    [tex]e^{-\infty}[/tex] = 0
    [tex]e^{0}[/tex] = 1

    [tex]\frac{\alpha\beta}{2}e^{-\alpha|x|}[/tex] * (-1)(0-1)
    =[tex]\frac{\alpha\beta}{2}e^{-\alpha|x|}[/tex]

    Hopefully I did this right! :smile:
     

    Attached Files:

  7. Apr 4, 2008 #6
    You know you can make LaTeX show limits of integration: [tex]\int_{-\infty}^{\infty}f(x)\,dx[/tex], etc.
     
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