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I have to calculate this integral $\int \frac{x}{\sin x}\,{\rm d}x$. Is there any way to evaluate this?

Thanks.

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  • $\begingroup$ Wolframalpha shows one in terms of the polylogarithm function $\endgroup$ – egreg Jan 26 '16 at 10:50
  • $\begingroup$ The key idea is to use a substitution $e^{ix}=z$. The resulting integral is easily expressed in terms of dilogarithms after performing an integration by parts. $\endgroup$ – tired Jan 26 '16 at 11:29
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This integral doesn't have a closed-form solution.

See here Wolfram Alpha

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  • $\begingroup$ How can I recognize this type of integrals ? $\endgroup$ – Nir Movshovitz Jan 26 '16 at 10:58
  • $\begingroup$ @NirMovshovitz If you really want to do so, you should keep several kinds of special integral in mind. However, in my opinion,wasting your time on that is meaningless. $\endgroup$ – Yijun Yuan Jan 26 '16 at 11:02
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This function's primitive is not expressable in terms of elementary functions. Basically, you're integrating $\frac{1}{\frac{\sin(x)}{x}}$

See here for a detailed explanation

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The integrand does not admit a closed form antiderivative. See Liouville's theorem and the Risch algorithm for more information. However, its definite counterpart evaluated over $\bigg(0,~\dfrac\pi2\bigg)$ yields twice the value of Catalan's constant as a result.

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