I integrate by part I assume $dx=dv$ and $\ln(\sin x)= u$ or
I compute by Maple but its answer wasn't clearly (Maple answer: xln(1-exp((2I)x))+xln(sin(x))+(1/2*I)x^2+(1/2*I)*polylog(2, e((2*I)*x)))
thanks for any hints
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doraemonpaul
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M.H
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Your formatting has gone astray. Can you please correct? Regards – Amzoti Jan 16 '13 at 20:05
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Do you have bounds? – NeverBeenHere Jan 16 '13 at 20:24
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I pasted Integral(log(sin(x)),x) into wolframalpha.com to get a result. The solution contained something called polylogarithm. – miracle173 Jan 16 '13 at 20:34
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whats polylogarithm ? how wolframalpha.com compute it? – M.H Jan 16 '13 at 20:41
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http://www.wolframalpha.com/ – miracle173 Jan 16 '13 at 20:48
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According to Mathematica the answer is $$ -x \text{Log}\left[1-e^{2 i x}\right]+x \text{Log}[\text{Sin}[x]]+\frac{1}{2} i \left(x^2+\text{PolyLog}\left[2,e^{2 i x}\right]\right) $$ Where PolyLog is Polylogarithm function. Of course this is a non elementary integral. But for certain bounds, you find that the integral converges to some interesting values. – S L Jan 16 '13 at 21:04
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If you know about Clausen function, the results can be much simpler. – doraemonpaul Jan 19 '14 at 18:38
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You posted an image. Please use latex and make the post searchable... Thanks. – Umberto Jan 19 '14 at 20:35