$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}}(2 \operatorname{cosec} x)^{17} dx$
Let $e ^{ u }+ e ^{- u }=2 \operatorname{cosec} x , x =\frac{\pi}{4} $
$\Rightarrow u =\ln (1+\sqrt{2}), x =\frac{\pi}{2}$
$ \Rightarrow u =0$
$\Rightarrow \operatorname{cosec} x+\cot x=e^{u}$
and $\operatorname{cosec} x-\cot x=e^{-u} $
$\Rightarrow \cot x=\frac{e^{u}-e^{-u}}{2}$
$\left(e^{u}-e^{-u}\right) d x=-2 \operatorname{cosec} x \cot x d x$
$\Rightarrow -\int\left( e ^{ u }+ e ^{- u }\right)^{17} \frac{\left( e ^{ u }- e ^{- u }\right)}{2 \operatorname{cosec} x \cot x } du$
$=-2 \int\limits_{\ln (1+\sqrt{2})}^{0}\left( e ^{ u }+ e ^{- u }\right)^{16} du$
$=\int\limits_{0}^{\ln (1+\sqrt{2})} 2\left( e ^{ u }+ e ^{- u }\right)^{16} du$