5.5 The Substitution Rule/33

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int {\sqrt {\cot(x)}}\csc ^{2}{(x)}dx}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} u &= \cot{(x)} \\[2ex] du &= -csc^2{(x)}dx \\[2ex] dx &= \frac{du}{-csc^2{(x)}} \\[2ex] \end{align} }

${\begin{aligned}&=\int {\sqrt {u}}\csc ^{2}{(x)}{\frac {du}{-csc{(x)}}}&=-\int {({\sqrt {u}})}du\\[2ex]&=\int (u^{\frac {1}{2}})du\\[2ex]&=-{\frac {2}{3}}u+c\\[2ex]&=-{\frac {2}{3}}(\cot {(x)})^{\frac {3}{2}}+c\end{aligned}}$