Question

See attached image for problem (evaluate the definite integral)

Answer 1

`\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 (\pi)/(3) - \tan (\pi)/(6) \right) +8 \left( \csc (\pi)/(3) - \csc (\pi)/(6)\right)\\\\\\`

`=2\left( \sqrt 3-\frac 1{\sqrt 3} \right) +8 \left( \frac 2{\sqrt 3} - 2 \right)\\\\\\=2\left( (2)/(\sqrt 3) \right) +8 \left((2-2\sqrt 3)/(\sqrt 3)\right)\\\\\\=\frac 4{\sqrt 3} + 16 \left((1-\sqrt 3)/(\sqrt 3)\right)\\\\\\=(4)/(\sqrt 3) \left( 1 + 4-4\sqrt 3 \right)\\\\\\=\frac 4{\sqrt 3}\left(5-4\sqrt 3\right)\\\\\\=(4)/(3)\left(5\sqrt 3-12\right)`