$I=$ $\int {\sqrt {1 + 2\cot x\csc ecx + 2{{\cot }^2}x} } \cdot dx$
$ \Rightarrow \quad I = \int {\sqrt {\frac{{{{\sin }^2}x + 2\cos x + 2{{\cos }^2}x}}{{{{\sin }^2}x}}} } \cdot dx$
$\Rightarrow 1=\int \frac{\sqrt{1}+2 \cos x+\cos ^{2} x}{\sin x} \cdot d x$
$ \Rightarrow \quad 1 = \int {\left| {\frac{{1 + \cos x}}{{\sin x}}} \right|} \cdot dx$
$ \Rightarrow \quad I = \int | \csc ecx + \cot x| \cdot dx$
$ \Rightarrow \quad {\rm{I}} = \log |\csc ecx - \cot x|$
$ + \log |\sin x| + {C_1}$
$\Rightarrow \quad I=\log |1-\cos x|+C_{1}$
$\Rightarrow \quad I=\log \left|2 \sin ^{2} \frac{x}{2}\right|+C_{1}$
$\Rightarrow \quad I=\log \left|\sin ^{2} \frac{x}{2}\right|+\log 2+C_{1}$
$\Rightarrow \quad I=2 \log \left|\sin \frac{x}{2}\right|+C$
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