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] \end{align} }

${\begin{aligned}\int {\frac {\cos(x)}{\sin(x)}}dx&=\int {\frac {1}{\sin(x)}}\cos(x)\;dx=\int {\frac {1}{\sin(x)}}(\cos(x)\;dx)\\[2ex]&=\int {\frac {1}{u}}(du)\\[2ex]{\text{Note: }}\int {\frac {1}{x}}dx=ln(x)+C\\[2ex]&=\left|ln(u)\right|+C\\[2ex]&=\left|ln(\sin(x))\right|+C\\[2ex]\end{aligned}}$