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I recently encountered an integral of the form: $$\int{\frac{\log(a+bx+\sqrt{x^2+c})}{x}}dx$$ The result involves the dilogarithm function, but I was wondering if there is a fast way of showing that the integral cannot be expressed in terms of elementary functions. I am aware of the existence of the Risch Algorithm, but I thought there may be a more intuitive way.

Is there such a method?

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    $\begingroup$ $\int\frac{\log(a+bx)}{x}\mathrm dx$ is easily transformed so that it looks like the definition of the dilogarithm, but I'm not sure that what you have is as amenable to such recognition... $\endgroup$ Jul 13, 2012 at 16:46
  • $\begingroup$ @J.M. - I realize that, but say I never heard of the dilogarithm, how would I know that this integral cannot be expressed in terms of elementary functions? "You can't" would also be a valid answer I guess.. $\endgroup$ Jul 13, 2012 at 16:50

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I can refer you to this paper : http://www.claymath.org/programs/outreach/academy/LectureNotes05/Conrad.pdf.

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