∫ 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 x = d u − c s c 2 ( x ) {\displaystyle {\begin{aligned}u&=\cot {(x)}\\[2ex]du&=-csc^{2}{(x)}dx\\[2ex]dx&={\frac {du}{-csc^{2}{(x)}}}\\[2ex]\end{aligned}}}
= ∫ u csc 2 ( x ) d u − c s c ( x ) = − ∫ ( 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}}\csc ^{2}{(x)}{\frac {du}{-csc{(x)}}}&=-\int {({\sqrt {u}})}du\\[2ex]&=\int (u^{\frac {1}{2}})du\\[2ex]&=-{\frac {2}{3}}u+c&=-{\frac {2}{3}}(\cot {(x)}^{\frac {3}{2}}+c\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=\cot {(x)}\\[2ex]du&=-csc^{2}{(x)}dx\\[2ex]dx&={\frac {du}{-csc^{2}{(x)}}}\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c63e3a61bf19f666a381da3e47331737ac9a503f)
![{\displaystyle {\begin{aligned}&=\int {\sqrt {u}}\csc ^{2}{(x)}{\frac {du}{-csc{(x)}}}&=-\int {({\sqrt {u}})}du\\[2ex]&=\int (u^{\frac {1}{2}})du\\[2ex]&=-{\frac {2}{3}}u+c&=-{\frac {2}{3}}(\cot {(x)}^{\frac {3}{2}}+c\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/a00a5f8dc562607f75308b838c0e03c5aad8031d)