Considering integrals of the form
$$\int_{0}^{a}\frac x{\sin(x)}\,dx \quad a\in\mathbb{R}$$
I was wondering when this integral has a closed form expression in terms of special functions if needed, instead of just a numerical approximation.
I already have evaluated it for $\pi\over2$ yielding:
$$\int_{0}^{\frac{\pi}{2}}\frac{x}{\sin(x)}\,dx=2G$$
Where $G$ is Catalans constant.
Anyone know of any else?
From the tables, there is no closed expression. As all integrals of functions involving rationals of trignonometrics, by the inversion of the simple $\arctan$ series for the $\tan$, the coefficients are trivial factors of $(-1)^n, 2^n, n!$ and Bernoulli numbers.
$$\int x \ \text{csc}(x) \ dx = x + 2 \ \sum _{k=1}^n \frac{ (-1)^{k+1} \left(2^{2 k-1}-1\right) \ B_{2 k} } {(2 k+1)!} \ \ x^{2 k+1}$$