证明：
$$int_{0}^{frac{pi}{2}}ln (1+cos x)dx=-frac{pi}{2}ln 2 +int_{0}^{frac{pi}{2}}frac{x}{sin x}dx$$
Proof.
egin{align*}
int_{0}^{frac{pi}{2}}ln (1+cos x) dx &=int_{0}^{frac{pi}{2}}ln(sin x (csc x + cot x))dx\
&=int_{0}^{frac{pi}{2}} ln sin x dx +int_{0}^{frac{pi}{2}}ln (csc x +cot x)dx\
&:=I_{1}+I_{2}
end{align*}
计算$I_{1}$和$I_{2}$
egin{align*}
int_{0}^{frac{pi}{2}}ln sin x dx+int_{0}^{frac{pi}{2}}ln cos x dx &=int_{0}^{frac{pi}{2}}ln frac{sin 2x}{2}dx\
&=-frac{pi ln 2}{2}+frac{1}{2}int_{0}^{pi}ln sin x dx\
&=-frac{pi ln 2}{2}+int_{0}^{frac{pi}{2}}ln cos x dx
end{align*}
从而 $I_{1}=-frac{pi ln 2}{2}$, $I_{2}$分部积分处理即可。