∫ cos ( x ) sin 2 ( x ) d x {\displaystyle \int {\frac {\cos {(x)}}{\sin ^{2}{(x)}}}dx}
u = sin ( x ) d u = cos ( x ) d x {\displaystyle {\begin{aligned}u&=\sin {(x)}\\[2ex]du&=\cos {(x)}dx\end{aligned}}}
∫ cos ( x ) sin 2 ( x ) d x = ∫ 1 u 2 d u = ∫ u − 2 d u = u − 1 + C = − 1 sin ( x ) + C {\displaystyle {\begin{aligned}\int {\frac {\cos {(x)}}{\sin ^{2}{(x)}}}dx&=\int {\frac {1}{u^{2}}}du=\int u^{-2}du\\[2ex]&=u^{-1}+C\\[2ex]&={\frac {-1}{\sin {(x)}}}+C\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=\sin {(x)}\\[2ex]du&=\cos {(x)}dx\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c9371c8d0e54e3be6eb14d031c53b463708cc897)
![{\displaystyle {\begin{aligned}\int {\frac {\cos {(x)}}{\sin ^{2}{(x)}}}dx&=\int {\frac {1}{u^{2}}}du=\int u^{-2}du\\[2ex]&=u^{-1}+C\\[2ex]&={\frac {-1}{\sin {(x)}}}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/01ea9bf3de6b2dbd1fe80ef10e3d5c61beae7f8d)