<s>[INST] How to find $\int_{0}^{\pi/2} \log ({1+\cos x}) dx$ using real-variable methods? How do you find the value of this integral, using real methods?
$$I=\displaystyle\int_{0}^{\pi/2} \log ({1+\cos x}) dx$$
The answer is $2C-\dfrac{\pi}{2}\log {2}$ where $C$ is Catalan's constant.
 [/INST] $$
\begin{aligned}
\int_{0}^{\frac{\pi}{2}} \ln (1+\cos x) d x &=\int_{0}^{\frac{\pi}{2}} \ln \left(2 \cos ^{2} \frac{x}{2}\right) d x \\
&=\int_{0}^{\frac{\pi}{2}} \ln 2 d x+2 \int_{0}^{\frac{\pi}{2}} \ln \left(\cos \frac{x}{2}\right) d x \\
&=\frac{\pi}{2} \ln 2+4 \int_{0}^{\frac{\pi}{4}} \ln (\cos x) d x \\
&=\frac{\pi}{2} \ln 2+4\left(-\frac{\pi}{2} \ln 2+\frac{G}{2}\right)
\end{aligned}
$$ where the last integral comes from my post:
$\int_{0}^{\frac{\pi}{4}} \ln (\cos x) d x=-\frac{\pi}{4} \ln 2+\frac{G}{2}$.
Hence we can conclude that
$$
\boxed{\int_{0}^{\frac{\pi}{2}} \ln (1+\cos x) d x= \frac{G}{2}-\frac{\pi}{2} \ln 2,}
$$
where G is the Catalan’s constant.
</s>