Let [ t ] denote the greatest integer ≤ t . Then 2 π ∫ π 6 5 π 6 8 cosec x - 5…

Let,

$I={\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}\left(8\left[\mathrm{cosec}x\right]-5\left[\cot x\right]\right)dx$

$\Rightarrow I=8 {\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}\left[\mathrm{cosec}x\right]dx-5{\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}\left[\cot x\right]dx$

$\Rightarrow I=8I_1-5I_2$, where $I_1=8 {\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}\left[\mathrm{cosec}x\right]dx \& I_2=5{\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}\left[\cot x\right]dx$

Now solving,

$I_1={\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}\left[\mathrm{cosec}x\right]dx$

$\Rightarrow I_1={\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}1dx=\frac{2\pi }{3}$ 

As when $x\in \left(\frac{\mathrm{\pi }}{6},\frac{5\mathrm{\pi }}{6}\right), \mathrm{cosec}x\in [1,2)$,

So, $\left[\mathrm{cosec}x\right]=1$

Now solving, 

$I_2={\int }_{\frac{\pi }{6}}^{\frac{5\pi }{6}}\left[\cot x\right]dx$

$\Rightarrow I_2={\int }_{\frac{\pi }{6}}^{\frac{\pi }{4}}1dx+{\int }_{\frac{\pi }{4}}^{\frac{\pi }{2}}0dx+{\int }_{\frac{\pi }{2}}^{\frac{3\pi }{4}}\left(-1\right)dx+{\int }_{\frac{3\pi }{4}}^{\frac{5\pi }{6}}\left(-2\right)dx$

$\Rightarrow I_2=\left(\frac{\pi }{4}-\frac{\pi }{6}\right)-\left(\frac{3\pi }{4}-\frac{\pi }{2}\right)-2\left(\frac{5\pi }{6}-\frac{3\pi }{4}\right)$

$\Rightarrow I_2=-\frac{\pi }{3}$

Required value will be,

$\frac{2}{\pi }I=\frac{2}{\pi }\left[8\times \frac{2\pi }{3}+5\times \frac{\pi }{3}\right]=14$