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This thing is killing me Integration Help

  1. Oct 9, 2011 #1
    This thing is killing me!!!! Integration Help

    So I was doing some research and came across the following indefinite integral:

    ∫tan(x)ln(x) dx

    where the domain of x is the complex plane so this can be re-written as:

    ∫tan(z)ln(z) dz...

    So I began solving the problem like so:

    ∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln(cos(x))/x dx (integration by parts)

    ∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + e^-ix)/2)/x dx (exponential definition of cos)

    ∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + e^-ix))/x dx - ln(2)ln(x) + C (properties of logarithms)

    ∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix)^2 + 1)/x dx - ix - ln(2)ln(x) + C (property of logarithms along with combining terms in the initial fraction)

    ∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + i)(e^ix - i))/x dx - ix - ln(2)ln(x) + C (expanding the sum of squares)

    Which leaves us with this as the remaining problem:

    ∫tan(x)ln(x) dx = -ln(cos(x))ln(x) + ∫ln((e^ix + i))/x dx + ∫ln((e^ix - i))/x dx - ix - ln(2)ln(x) + C

    So how on earth do you solve these two problems:

    ∫ln((e^ix + i)/x dx

    ∫ln(e^ix - i))/x dx

    I tried using Wolfram Mathematica but it could not integrate this problem and I can't imagine how a discrete method would work on this.
  2. jcsd
  3. Oct 9, 2011 #2
    Re: This thing is killing me!!!! Integration Help

    Maybe it doesn't have an elementary antiderivative. Can still do a nice job integrating it numerically as long as you go around the poles and branch point.
  4. Oct 9, 2011 #3
    Re: This thing is killing me!!!! Integration Help

    I belive strongly that this integral does not have a elementary antiderivative.

    Although is it possible to find an exact value for the integral below? (no approximations)

    [tex] I = \int_{0}^{1} \tan(x) \ln(x) \, dx [/tex]
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