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The integral of x/lnx

  1. Jun 1, 2010 #1
    For some reason, I can not seem to evaluate this integral.

    [tex]\int \frac{x}{ln(x)} dx[/tex]

    When I plug it into maple it says the solution is

    [tex]\frac{1}{2} \frac{x^2}{ln(x)}[/tex]

    I cannot seem to get it for some reason! I have tried every integration technique in the book. :\
  2. jcsd
  3. Jun 1, 2010 #2


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    That is not a correct result, as you can check by differentiating. It looks like you accidentally wrote log(y) or something which mathematica treated as a constant, and so it gave you the integral of x. The real result involves the exponential integral,


  4. Jun 1, 2010 #3
    You were right, I forgot to put the parenthesis on the natural logarithm. I haven't learned what an exponential integral is yet. I am currently in Differential Equations but "Ei" doesn't look familiar. :\
  5. Jun 1, 2010 #4


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    The Exponential Integral funtion, [itex]\mbox{Ei}(x)[/itex], is a special function defined by

    [tex]\mbox{Ei}(x) = \int_{-\infty}^x dt~\frac{e^t}{t}[/tex]

    To be fair, this is just obtained by making a change of variables on your original integral; however, this is a special function that should be available in mathematica or matlab, so it is a standard-ish form you can use.
  6. Jun 1, 2010 #5
    I did get that form quite a few times when I was trying to solve it. But I stopped at that point because it seemed like a dead end. So would that mean that the solution to the integral is an integral? Would this suggest that there is no actual solution to it? I guess you would use approximation techniques at that point to evaluate it. Am I correct with this assumption?
  7. Jun 2, 2010 #6
    I haven't looked at it closely, but it just looks like the integration by parts using u = x and setting dv as the logarithmic integral.
  8. Jun 2, 2010 #7


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    Your integral is one that can't be expressed in elementary functions. However, while the exponential integral function is still defined as an integral, it is a standard function that is recognized by many computer math systems, and so this form is more useful than the original integral form. The programs will use a variety of techniques to evaluate the function for different regimes of x, but I do not the specific details.
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