5.5 The Substitution Rule/33

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

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

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