∫ cot ( x ) d x = ∫ cos ( x ) sin ( x ) d x {\displaystyle \int \cot(x)dx=\int {\frac {\cos(x)}{\sin(x)}}dx}
u = sin ( x ) d u = cos ( x ) d x {\displaystyle {\begin{aligned}u&=\sin(x)\\[2ex]du&=\cos(x)\;dx\\[2ex]\end{aligned}}}
∫ cos ( x ) sin ( x ) d x = ∫ 1 sin ( x ) cos ( x ) d x = ∫ 1 sin ( x ) ( cos ( x ) d x ) = ∫ 1 u ( d u ) Note: ∫ 1 x d x = l n ( x ) + C = | l n ( u ) | + C = 2 3 ( 3 a x + b x 3 ) 1 / 2 + C = 2 3 3 a x + b x 3 + C {\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]&={\frac {2}{3}}(3ax+bx^{3})^{1/2}+C\\[2ex]&={\frac {2}{3}}{\sqrt {3ax+bx^{3}}}+C\end{aligned}}}
![{\displaystyle {\begin{aligned}u&=\sin(x)\\[2ex]du&=\cos(x)\;dx\\[2ex]\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/d86146c5ab9052b4f1c419e803482f36424c663c)
![{\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]&={\frac {2}{3}}(3ax+bx^{3})^{1/2}+C\\[2ex]&={\frac {2}{3}}{\sqrt {3ax+bx^{3}}}+C\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/3802cd9245f333af7e6e86f33036120b2440d01e)