5.5 The Substitution Rule/33

$\int {\sqrt {\cot(x)}}\csc ^{2}{(x)}dx$

${\begin{aligned}u&=\cot {(x)}\\[2ex]du&=-csc^{2}{(x)}dx\\[2ex]dx&={\frac {du}{-csc^{2}{(x)}}}\\[2ex]\end{aligned}}$

${\begin{aligned}&=\int {\sqrt {u}}\csc ^{2}{(x)}{\frac {du}{-csc{(x)}}}&=-\int {({\sqrt {u}})}du\\[2ex]&=\int (u^{\frac {1}{2}})du\\[2ex]&=-{\frac {2}{3}}u+c&=-{\frac {2}{3}}(\cot {(x)}^{\frac {3}{2}}+c\end{aligned}}$