∫ 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 = c s c 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/e6b295773530d29c8f9493965b5e798610865770)
![{\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/95d01179a4543641434ba89283169e3ff21c81e8)