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Integration using an Abel transform

  1. May 10, 2013 #1
    1. The problem statement, all variables and given/known data

    Find the following integral:

    2. Relevant equations

    [tex]\int \frac{e^{x}}{\sqrt{(1+e^{2x})(1-e^{4x})}}dx[/tex]

    3. The attempt at a solution

    I changed the integral to: [tex]\int \frac{e^{x}}{(1+e^{2x})\sqrt{(1-e^{2x})}}dx[/tex]
    The let u=e^x
    The integral becomes: [tex]\int \frac{du}{(1+u^{2})\sqrt{(1-u^{2})}}[/tex]
    I can do this the long way, such as on wolfram alpha but I want to use an Abel transform so let [tex]u=\sqrt{1-u^{2}}'[/tex]

    [tex]\sqrt{1-u^{2}}'=-\frac{u}{\sqrt{1-u^2}} \therefore v^{2}=\frac{u^{2}}{1-u^{2}}[/tex]

    [tex]du=\frac{dv}{\sqrt{1-u^{2}}}[/tex]

    The integral becomes: [tex]\int \frac{dv}{1-u^{4}}[/tex]

    I need to somehow get rid off the u and get the integral in terms of v so how can I do that?
     
  2. jcsd
  3. May 10, 2013 #2

    haruspex

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    u2 = 1 - v2, no?
     
  4. May 10, 2013 #3
    How do you get that?
     
  5. May 10, 2013 #4

    haruspex

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    Maybe I misunderstood your substitutions. This doesn't seem to be consistent:
    Did you mean [tex]v=\sqrt{1-u^{2}}'[/tex]? If so, u2 = v2/(1+v2)
     
  6. May 10, 2013 #5
    Thanks :)
     
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