Twin integrals, would like to know them precise.
$$ \begin{align} \int_0^{\pi/2}\log(1-\cos(x))\,\mathrm{d}x &=\int_0^{\pi/2}\left[\log(1-e^{ix})+\log(1-e^{-ix})-\log(2)\right]\,\mathrm{d}x\\ &=-\frac\pi2\log(2)-2\int_0^{\pi/2}\sum_{k=1}^\infty\frac{\cos(kx)}{k}\,\mathrm{d}x\\ &=-\frac\pi2\log(2)-2\sum_{k=1}^\infty\frac{\sin(k\pi/2)}{k^2}\\ &=-\frac\pi2\log(2)-2\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^2}\\ &=-\frac\pi2\log(2)-2\mathrm{G} \end{align} $$ where $\mathrm{G}$ is Catalan's Constant.