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\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
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\begin{align}
I & = \color{#f00}{-\int_{\pi}^{2\pi}
{\sin\pars{x} \over \sin\pars{x} - 1}\,\dd x} =
\int_{\pi}^{2\pi}{\sin\pars{x} + \sin^{2}\pars{x} \over \cos^{2}\pars{x}}\,\dd x
\\[5mm] & =
\int_{\pi}^{2\pi}{\sin\pars{x} \over \cos^{2}\pars{x}}\,\dd x
+ \int_{\pi}^{2\pi}\tan^{2}\pars{x}\,\dd x =
\int_{\pi}^{2\pi}{\sin\pars{x} \over \cos^{2}\pars{x}}\,\dd x
+ \int_{\pi}^{2\pi}\sec^{2}\pars{x}\,\dd x - \int_{\pi}^{2\pi}\dd x
\\[5mm] &=
\left.{1 \over \cos\pars{x}} + \tan\pars{x} - x\,\right\vert_{\ \pi}^{\ 2\pi} =
\color{#f00}{2 - \pi}
\end{align}