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Need hints on how to integrate this

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  • #1
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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!
 

Answers and Replies

  • #2
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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:
  • #3
Gib Z
Homework Helper
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To where you are now, evaluating the integral should be easy, noting that the remaining integrand is an even function.
 
  • #4
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Figured it out. Thanks!
 
  • #5
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=[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:
 

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

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