How to find the mean value of this function? $-\ln(\cos x)$

Question

I wonder whether it is possible to find the mean value of the function $-\ln(\cos x)$? Notice, the function is not bounded.

Answer 1

The function $-\ln(\cos x)$ is not defined everywhere, but we may be able to find the mean on a defined interval, note it is not always possible. In this case the mean is well defined.

The mean on $\displaystyle \left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ would be $\displaystyle \lim_{a \to \frac{\pi}{2}}\frac{1}{2a}\int_{-a}^a-\ln(\cos x)\,dx=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}-\ln(\cos x)\,dx = \log(2)$

Answer 2

$$\frac{2}{\pi}\int_{0}^{\pi/2}\log\cos(x)\,dx=\frac{2}{\pi}\int_{0}^{\pi/2}\log\sin(x)\,dx=\frac{2}{\pi}\int_{0}^{1}\frac{\log(x)}{\sqrt{1-x^2}}\,dx=\frac{1}{2\pi}\int_{0}^{1}\frac{\log x}{\sqrt{x(1-x)}}\,dx $$ equals $-\log 2$, also by symmetry tricks.