∫ cot ( x ) csc 2 ( x ) d x {\displaystyle \int {\sqrt {\cot(x)}}\csc ^{2}{(x)}dx}
u = cot ( x ) d u = − c s c 2 ( x ) d x − d u = csc 2 ( x ) d x {\displaystyle {\begin{aligned}u&=\cot {(x)}\\[2ex]du&=-csc^{2}{(x)}dx\\[2ex]-du&=\csc ^{2}{(x)}dx\\[2ex]\end{aligned}}}
= − ∫ ( u ) d u = ∫ ( u 1 2 ) d u = − 2 3 u + C = − 2 3 ( cot ( x ) ) 3 2 + C {\displaystyle {\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}}}
![{\displaystyle {\begin{aligned}u&=\cot {(x)}\\[2ex]du&=-csc^{2}{(x)}dx\\[2ex]-du&=\csc ^{2}{(x)}dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a7aa6d29473fde86c1f7ab0384be25562c694d8c)
![{\displaystyle {\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}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d36892f621a96322f9cf2f04eab90f564fbbbfb4)