Background:
I recently had to compute the following Riemann-Stieltjes integral for a homework assignment:
$$\int_\limits{-\pi}^{\pi}\cos(x)d|\sin(x)|=0$$
It occured to me that this looks similar to the symmetric Riemann integral $$\int_\limits{-\pi}^{\pi}\cos(x)dx=2\int_\limits{0}^{\pi}\cos(x)dx=0$$ which we know is 0 without computing because $\cos(x)$ is symmetric about $\pi/2$.
Question:
Is there any theory to know the answer (and to identify) symmetric Riemann-Stieltjes integrals?