$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
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\newcommand{\ic}{\mathrm{i}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
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\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
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\begin{align}
I & \equiv \int_{0}^{\pi}{x\sin\pars{x} \over a + b\cos^{2}\pars{x}}\,\dd x =
\int_{-\pi/2}^{\pi/2}{\pars{x + \pi/2}\cos\pars{x} \over a + b\sin^{2}\pars{x}}\,\dd x
\\[5mm] & =
\pi\int_{0}^{\pi/2}{\cos\pars{x} \over a + b\sin^{2}\pars{x}}\,\dd x =
\pi\int_{0}^{1}{\dd x \over bx^{2} + a}
\\[5mm] & =
\pi\,{1 \over a}\,\root{a \over b}\int_{0}^{1}{\root{b/a}\dd x \over \pars{\root{b/a}x}^{2} + 1} =
{\pi \over \root{ab}}\int_{0}^{\root{b/a}}{\dd x \over x^{2} + 1}
\\[5mm] & =
\bbx{{\pi \over \root{ab}}\,\arctan\pars{\root{b \over a}}}
\end{align}