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Trying to figure out integral with infitnite limites

  1. Oct 31, 2009 #1
    1. The problem statement, all variables and given/known data

    [tex]\int^{\infty}_{-\infty}(1/(a^{4}+(x-x_{0})^{4}))dx[/tex]

    2. Relevant equations



    3. The attempt at a solution

    i let [tex]u = (x-x_{0})^{4}[/tex]

    but have no idea what to go from there
     
    Last edited: Oct 31, 2009
  2. jcsd
  3. Oct 31, 2009 #2

    Dick

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    I would use u=(x-x0)/a and factor out the a^4. That gives you 1/(1+u^4). (1+u^4)=(u^2-sqrt(2)u+1)*(u^2+sqrt(2)u+1). Use partial fractions on that. It's not an easy integral, but it can be done.
     
  4. Oct 31, 2009 #3
    awesome thanks! got [tex]\pi*\sqrt{2}/2[/tex]
     
  5. Oct 31, 2009 #4

    Dick

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    Good job. Don't forget to put the 'a' factor back in again.
     
  6. Oct 31, 2009 #5
    As Dick said, firstly let [tex]
    u=\frac{x-x_{0}}{a} [/tex] then [tex] du= adx [/tex] and now integral becomes

    [tex]
    \frac{1}{a^{5}}\int^{\infty}_{-\infty}\frac{1}{1+u^{4}}du
    [/tex]
    Secondly, by letting [tex] u=e^{i\theta} [/tex] and [tex] du=i*{e}^{i\theta}d\theta[/tex]
    you can use "residue theorem". Yet, I forgot how can we apply here. After I remember, I'll post it
     
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