# Calculate the indefinite integral $\int \frac{x}{\sin x}\,{\rm d}x$

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

Thanks.

• Wolframalpha shows one in terms of the polylogarithm function – egreg Jan 26 '16 at 10:50
• 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. – tired Jan 26 '16 at 11:29

This function's primitive is not expressable in terms of elementary functions. Basically, you're integrating $\frac{1}{\frac{\sin(x)}{x}}$
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.