证明定积分等式

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.

\begin{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$I_{2}$

\begin{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。