[tex]\displaystyle \int^{\tfrac{ \pi} 3}_{\tfrac{\pi}6} \left[2\sec^2 (w) - 8\csc (w) \cot (w)\right]~dw\\\\\\=2\displaystyle \int^{\tfrac{ \pi} 3}_{\tfrac{\pi}6} \sec^2 (w)~ dw -8\displaystyle \int^{\tfrac{ \pi} 3}_{\tfrac{\pi}6} \csc (w) \cot(w) ~ dw\\\\\\=2 \biggr[\tan (w) \biggr]^{\tfrac{\pi}3}_{\tfrac{\pi}6} +8\biggr[\csc (w) \biggr]^{\tfrac{\pi}3}_{\tfrac{\pi}6}\\\\\\=2\left( \tan \dfrac{\pi}{3} - \tan \dfrac{\pi}{6} \right) +8 \left( \csc \dfrac{\pi}{3} - \csc \dfrac{\pi}{6}\right)\\\\\\[/tex]

[tex]=2\left( \sqrt 3-\dfrac 1{\sqrt 3} \right) +8 \left( \dfrac 2{\sqrt 3} - 2 \right)\\\\\\=2\left( \dfrac{2}{\sqrt 3} \right) +8 \left(\dfrac{2-2\sqrt 3}{\sqrt 3}\right)\\\\\\=\dfrac 4{\sqrt 3} + 16 \left(\dfrac{1-\sqrt 3}{\sqrt 3}\right)\\\\\\=\dfrac{4}{\sqrt 3} \left( 1 + 4-4\sqrt 3 \right)\\\\\\=\dfrac 4{\sqrt 3}\left(5-4\sqrt 3\right)\\\\\\=\dfrac{4}{3}\left(5\sqrt 3-12\right)[/tex]