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.

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。