5.5 The Substitution Rule/37

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

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

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