5.5 The Substitution Rule/31

$\int {\frac {\cos {(x)}}{\sin ^{2}{(x)}}}dx$

${\begin{aligned}u&=\sin {(x)}\\[2ex]du&=\cos {(x)}dx\end{aligned}}$

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