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Explain how to compute this integral?

  1. Nov 24, 2008 #1
    Can someone please explain how to compute this integral? This is not for school; I just came across it and I'm not sure what to do. Parts and various identities didn't help. [tex]\int[/tex] x/tan(x) dx
     
  2. jcsd
  3. Nov 24, 2008 #2
    Re: Integration

    Parts should help. Let u(x) = x and let dv(x) = 1/tan(x). Then v(x) = ln|sin(x)| + C and du(x) = 1.
     
  4. Nov 24, 2008 #3
    Re: Integration

    This only results in having to integrate ln|sin(x)|, which does not help. I'm pretty sure parts is not the way to go.
     
  5. Nov 24, 2008 #4

    Hootenanny

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    Re: Integration

    Now use integration by parts again with u(x) = ln|sin(x)| and dv(x) = 1. (You'll find that you'll have to use integration by parts once again after this step).
     
    Last edited: Nov 24, 2008
  6. Nov 24, 2008 #5
    Re: Integration

    after the first integration by parts i get xln|sin(x)|-[tex]\int[/tex]ln|sin(x)|dx

    using parts again, with u(x)=ln|sin(x)|, i get xln|sin(x)|-xln|sin(x)|-[tex]\int[/tex]x/tanx dx

    this is exactly where i started.

    I entered this in an online integrator and got this as the answer:

    xln(1-e2ix)-1/2i(x2+Li2(e2ix)

    which does not make sense to me.
     
  7. Nov 25, 2008 #6

    Hootenanny

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    Re: Integration

    Hmm, indeed this isn't really helpful. Initially, I thought that we could evaluate this integral using recurrence methods. However, the result you obtained from the online integrator suggests that the anti-derivative cannot be written in terms of elementary functions.
    Li(x) the so-called logarithmic integral function and is a special function. See http://en.wikipedia.org/wiki/Logarithmic_integral_function for more information.
     
  8. Nov 25, 2008 #7

    gabbagabbahey

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    Re: Integration

    Actually, in this case [tex]L_i(z)[/tex] represents the PolyLogarithm Function....which seems to suggest that perhaps it can be integrated by writing tanx in terms of complex exponentials and using a complex analysis method such as method of residues. Although I haven't tried it yet myself, it seems like a reasonable approach given the form of the solution.
     
  9. Nov 25, 2008 #8
    Re: Integration

    Thank you. This seems more probable. Is there anyway you can show me how to actually integrate this function or suggest any sites that would show how to evaluate these type of integrals.
     
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