Evaluate $\int_0^{\pi/2}\log\cos(x)\,\mathrm{d}x$ How can you evaluate $$\int\limits_0^{\pi/2}\log\cos(x)\,\mathrm{d}x\;?$$

Here is an approach. Making the changes of variables $u=\cos x$ and $u^2=t$ in a row gives 

$$ I = \frac{1}{4}\int _{0}^{1}\!\,{\frac {\ln  \left( t \right) }{\sqrt {1-t}
\sqrt {t}}}{dt}
.$$

To evaluate the above integral, let's consider the following beta function 

$$ F = \int_{0}^{1} t^a (1-t)^b dt = \beta(a+1,b+1)=\frac{\Gamma(a+1)\Gamma(b+1)}{\Gamma(a+b+2)} .$$

Now, our integral follows from $F$ as

$$ I = \lim_{b\to -1/2}\lim_{a\to -1/2}F_a(a,b)=-\frac{\pi\ln 2}{2},\quad F_a =\frac{dF}{da}. $$

