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\begin{align}
I&=\color{#00f}{\large\int_{0}^{\pi}{\dd x \over 1 + \cos^2\pars{x}}}=
2\int_{0}^{\pi/2}{\dd x \over 1 + \sin^{2}\pars{x}}
=
2\int_{0}^{\pi/2}{\dd x \over 1 + \bracks{1 - \cos\pars{2x}}/2}
\\[3mm]&=
2\int_{0}^{\pi}{\dd x \over 3 - \cos\pars{x}}
=2\int_{0}^{\infty}{1 \over 3 - \pars{1 - t^{2}}/\pars{1 + t^{2}}}
\,{2\,\dd t \over 1 + t^{2}}
=4\int_{0}^{\infty}{1 \over 4t^{2} + 2}\,\dd t
\\[3mm]&=\root{2}\int_{0}^{\infty}{1 \over \pars{\root{2}t}^{2} + 1}\,\root{2}\,\dd t
=\color{#00f}{\large{\root{2} \over 2}\,\pi}
\end{align}
where $\ds{t \equiv \tan\pars{x \over 2}}$.
