5.5 The Substitution Rule/37

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

${\begin{aligned}u&=\sin(x)\\[2ex]du&=\cos(x)\;dx\\[2ex]\end{aligned}}$

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int {\frac {\cos(x)}{\sin(x)}}dx&=\int {\frac {1}{\sin(x)}}\cos(x)\;dx=\int {\frac {1}{\sin(x)}}(\cos(x)\;dx)\\[2ex]&=\int {\frac {1}{u}}(du)\\[2ex]{\text{Note: }}\int {\frac {1}{x}}dx=ln(x)+C\\[2ex]&=\left|ln(u)\right|+C\\[2ex]&=\left|ln(\sin(x))\right|+C\\[2ex]\end{aligned}}}