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I used the method of integration by parts (in many ways) to calculate $$\int \frac {\ln x}{1-x^2}\mathrm{d}x$$ But I'm still not successful. Is there any way of knowing that it is impossible to calculate this integral in terms of elementary functions?

When I used symbolic calculation software I got this $$\frac{1}{2}\left[{\rm{L}}{{\rm{i}}_2}(1 - x) + {\rm{L}}{{\rm{i}}_2}(1 - x) + \ln x \cdot \ln \,(x + 1)\right] + C$$ Where $Li_n$ is a polylogarithmic function.

In the first semester of the university they still do not teach functions such as Polylogarithmic, Beta, Bessel or error function, complex variable functions, and so on.

One only wants to express it as algebraic functions, or transcendent elementary functions.

mathsalomon
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  • I want to say maybe not, and first year students don't encounter integrals like this anyway. – Sean Roberson May 31 '17 at 18:10
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    All antiderivatives of a function are the same (within a constant) so if one antiderivative of that function is the one that your calculator gave you, then all of them are in that form. You're going to have to use the polylogarithm. – Franklin Pezzuti Dyer May 31 '17 at 18:14
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    "Is there a way of knowing that it is impossible to calculate this integral in terms of elementary functions". Yes, there is. – projectilemotion May 31 '17 at 18:18
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    A dilogarithm is a dilogarithm, you cannot turn it into something simpler. However, for some particular integration ranges the given integral can be computed through suitable symmetry tricks. – Jack D'Aurizio May 31 '17 at 20:04
  • Indeed, as I was thinking, the teacher who proposed this exercise did not put it in the section that corresponds to it – mathsalomon Jun 01 '17 at 22:04

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