5.5 The Substitution Rule/37

$\int \cot(x)dx=\int {\frac {\cos(x)}{\sin(x)}}dx$

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 &= \sin(x) \\[2ex] du &= (\cos(x))dx \\[2ex] \frac{1}{3}du &= (a+bx^2)dx \\[2ex] \end{align} }

${\begin{aligned}\int {\frac {a+bx^{2}}{\sqrt {3ax+bx^{3}}}}dx&=\int {\frac {1}{\sqrt {3ax+bx^{3}}}}(a+bx^{2})\;dx=\int {\frac {1}{\sqrt {3ax+bx^{3}}}}(a+bx^{2}\;dx)\ \\[2ex]&={\frac {1}{3}}\int {\frac {1}{\sqrt {u}}}(du)={\frac {1}{3}}\int u^{-1/2}du\\[2ex]&={\frac {1}{3}}{\frac {u^{\frac {1}{2}}}{\frac {1}{2}}}+C\\[2ex]&={\frac {2}{3}}(3ax+bx^{3})^{1/2}+C\\[2ex]&={\frac {2}{3}}{\sqrt {3ax+bx^{3}}}+C\end{aligned}}$