Mathematics - Definite Integration Question with Solution | TestHub

$\begin{array}{rl}& I={\int }_0^{\pi }\frac{8xdx}{4{\cos }^{2}x+{\sin }^{2}x}\\ & I={\int }_0^{\pi }\frac{8(\pi -x)dx}{4{\cos }^{2}x+{\sin }^{2}x}\end{array}$
$2\mathrm{I}=8\pi {\int }_0^{\pi }\frac{dx}{4{\cos }^{2}x+{\sin }^{2}x}$
$2\mathrm{I}=8\pi \times 2{\int }_0^{\pi /2}\frac{{\sec }^{2}x}{4+{\tan }^2\mathrm{x}}\mathrm{dx}$
$\begin{array}{rl}& \mathrm{I}=8\pi {\int }_0^{\infty }\frac{\mathrm{dt}}{4+{\mathrm{t}}^2}=8\pi \times \frac{1}{2}{\left({\tan }^{-1}\frac{\mathrm{t}}{2}\right]}_0^{\infty }\\ & =4\pi \times \frac{\pi }{2}=2{\pi }^2\end{array}$
option (1)