$3 \int \limits_{\pi / 3}^{\pi / 2} \frac{d x}{1+\cos x}$
$\int \limits_{\pi / 3}^{\pi / 2} \frac{d x}{1+\cos x}=\int \limits_{\pi / 3}^{\pi / 2} \frac{1-\cos x}{\sin ^2 x} d x$
$=\int \limits_{\pi / 3}^{\pi / 2}\left(\operatorname{cosec}{ }^2 x-\cot x \operatorname{cosec} x\right) d x$
$=(\operatorname{cosecx}-\cot x) \int \limits_{\pi / 3}^{\pi / 2}=(1)-\left(\frac{2}{\sqrt{3}}-\frac{1}{\sqrt{3}}\right)=1-\frac{1}{\sqrt{3}}$
$\int \limits_{\pi / 3}^{\pi / 2} \frac{d x}{\sin x(1+\cos x)}= \int \frac{d x}{(2 \tan x / 2)\left(1+1-\tan ^2 x / 2\right)}$
$=\int \frac{\left(1+\tan ^2 x / 2\right) \sec ^2 x / 2 d x}{2 \tan x / 2}$
$\tan x / 2= t \quad \sec x / 2 \frac{1}{2} dx = dt$
$\frac{1}{2} \int\left(\frac{1+ t ^2}{ t }\right) dt =\frac{1}{2}\left[\ell nt +\frac{ t ^2}{2}\right]_{\frac{1}{\sqrt{3}}}^1$
$=\frac{1}{2}\left[\left(0+\frac{1}{2}\right)-\left(\ln \frac{1}{\sqrt{3}}+\frac{1}{6}\right)\right]=\left(\frac{1}{3}+\ell n \sqrt{3}\right) \frac{1}{2}$
$=\left(\frac{1}{6}+\frac{1}{2} \ell n \sqrt{3}\right)$
$2\left(\frac{1}{6}+\frac{1}{2} \ell n \sqrt{3}\right)+3\left(1-\frac{1}{\sqrt{3}}\right)$
$=\frac{1}{3}+\ell n \sqrt{3}+3-\sqrt{3}=\frac{10}{3}+\ln \sqrt{3}-\sqrt{3}$
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