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How to compute the following integral? $$\int\log(\sin x)\,dx$$


Motivation: Since $\log(\sin x)'=\cot x$, the antiderivative $\int\log(\sin x)\,dx$ has the nice property $F''(x)=\cot x$. Can we find $F$ explicitly? Failing that, can we find the definite integral over one of intervals where $\log (\sin x)$ is defined?

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    I'm pretty sure this is an integral that can't be expressed in terms of elementary functions (that is, the functions of 1st-year calculus). See, for example, http://reference.wolfram.com/legacy/v5/TheMathematicaBook/AdvancedMathematicsInMathematica/Calculus/3.5.7.html about halfway down the page.2011-05-08
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    Yes, the dilogarithm seems to be required here...2011-05-08
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    @Kolya: Do you actually want to compute $\int_a^b {\log (\sin (x))\,{\rm d}x}$ for certain $a$ and $b$?2011-05-08
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    For $a=0$ and $b=\pi/2$ or $b=\pi$, for example...2011-05-08
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    @Kolya: You need to specify the limits. Note that the limits can only be between $(2n \pi, (2n+1) \pi)$.2011-05-08
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    @Sivaram: it's simpler (and you've no need to know about dilogs) with those limits, but you can do the indefinite integral...2011-05-08
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    Although this integral may cannot be expressed in elementary function, but it may can be expressed in series form. For example, ∫sin(sin x)dx and ∫cos(cos x)dx can both be evaluated in series form.2012-07-12

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