I = $\underset{0}{\overset{\mathit{\pi }}{\int }}$ $\frac{\mathit{x}\mathit{i}\mathit{s}\mathit{n}\mathit{x}}{1+\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{x}}$ dx ... (i)
I = $\underset{0}{\overset{\mathit{\pi }}{\int }}$ $\frac{\mathit{\pi }-\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{\pi }-\mathit{x}}{1+\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{x}\mathit{\pi }-\mathit{x}}$ dx
I = $\underset{0}{\overset{\mathit{\pi }}{\int }}$ $\frac{\mathit{\pi }-\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{1+\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{x}}$ dx
I = $\underset{0}{\overset{\mathit{\pi }}{\int }}$ $\frac{\mathit{\pi }\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{1+\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{x}}$ dx - $\underset{0}{\overset{\mathit{\pi }}{\int }}$ $\frac{\mathit{x}\mathit{s}\mathit{i}\mathit{n}\mathit{x}}{1+\mathit{c}\mathit{o}{\mathit{s}}^2\mathit{x}}$ dx ... (2)
Adding (1) and (2), we get:
2I = $\underset{1}{\overset{-1}{\int }}\frac{\mathit{\pi }-\mathit{d}\mathit{t}}{1+{\mathit{t}}^2}$
2I = - π $\underset{1}{\overset{-1}{\int }}$ $\left(\frac{1}{1+{\mathit{t}}^2}\right)$ dt
2I = - π ${\left|\mathit{t}\mathit{a}{\mathit{n}}^{-1}\mathit{t}\right|}_1^{-1}$
2I = π [$\mathit{t}\mathit{a}{\mathit{n}}^{-1}1-\mathit{t}\mathit{a}{\mathit{n}}^{-1}1$ - 1]
2I = π $\left(\frac{\mathit{\pi }}{4}-\left(-\frac{\mathit{\pi }}{4}\right)\right)$
2I = $\frac{{\mathit{\pi }}^2}{2}$
∴ I = $\frac{{\mathit{\pi }}^2}{4}$