0

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?

Angelo
  • 13,952
Person
  • 1,233
  • 3
  • 11

1 Answers1

0

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.

$$\int x \ \text{csc}(x) \ dx = x + 2 \ \sum _{k=1}^n \frac{ (-1)^{k+1} \left(2^{2 k-1}-1\right) \ B_{2 k} } {(2 k+1)!} \ \ x^{2 k+1}$$

Roland F
  • 5,122