JEE 2023 — Mathematics PYQ

\begin{aligned}
 & \mathrm{Let~I}= {\frac{2}{\pi}}\int_{\frac{\pi}{6}}^{5{\frac{\pi}{6}}}\{8[\cos x]-5[\cot x]\}dx \\
 & =\frac{2}{\pi}\Bigg[
\begin{matrix}
{\frac{5\pi}{6}} & {\frac{5\pi}{6}} \\
{8\int\limits_{
\begin{array}
{c}{\pi} \\
{6}
\end{array}}[\mathrm{cosec}x]dx-5\int\limits_{
\begin{array}
{c}{\pi} \\
{6}
\end{array}}[\cot x]dx} \\
\end{matrix}\Bigg] \\
 & =\frac{2}{\pi}\Bigg[8\int\limits_{\pi/6}^{5\pi/6}dx-5\Bigg\{\int\limits_{\pi/6}^{\pi/4}dx+\int\limits_{\pi/4}^{\pi/2}0.dx+\int\limits_{\pi/2}^{3\pi/4}(-1)dx+ \\
 & \left.+\int_{3\pi/4}^{5\pi/6}(-2)dx\right\} \\
 & =\frac{2}{\pi}\Bigg[8\times\Bigg(\frac{5\pi}{6}\frac{-\pi}{6}\Bigg)-5\Bigg\{\Bigg(\frac{\pi}{4}-\frac{\pi}{6}\Bigg)-\Bigg(\frac{3\pi}{4}-\frac{\pi}{2}\Bigg)\Bigg\} \\
 & -2\biggl(\frac{5\pi}{6}-\frac{3\pi}{4}\biggr)\biggr] \\
 & =\frac{2}{\pi}\biggl[\frac{16\pi}{3}+\frac{5\pi}{3}\biggr]=14
\end{aligned}