If we want to integrate the following:
$$\int_{-\pi}^{\pi}\frac{1}{2}\log(5 + 4 \cos(x))\,d\mu(x).$$
How can we determine the measure in a specific interval? and how can we know if the measure is symmetric or non-symmetric?
For example:
the measure $d\mu(x)= \dfrac{\pi}{\sinh(\pi)}\,d(e^{(x)})$ on the interval $[-\pi,\pi]$ is non-symmetric.