$$$\cot{\left(t \right)}$$$ 的积分

求$$$\int \cot{\left(t \right)}\, dt$$$。

将余切改写为 $$$\cot\left(t\right)=\frac{\cos\left(t\right)}{\sin\left(t\right)}$$$:

$${\color{red}{\int{\cot{\left(t \right)} d t}}} = {\color{red}{\int{\frac{\cos{\left(t \right)}}{\sin{\left(t \right)}} d t}}}$$

设$$$u=\sin{\left(t \right)}$$$。

则$$$du=\left(\sin{\left(t \right)}\right)^{\prime }dt = \cos{\left(t \right)} dt$$$ (步骤见»)，并有$$$\cos{\left(t \right)} dt = du$$$。

因此，

$${\color{red}{\int{\frac{\cos{\left(t \right)}}{\sin{\left(t \right)}} d t}}} = {\color{red}{\int{\frac{1}{u} d u}}}$$

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$${\color{red}{\int{\frac{1}{u} d u}}} = {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回忆一下 $$$u=\sin{\left(t \right)}$$$:

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(t \right)}}}}\right| \right)}$$

因此，

$$\int{\cot{\left(t \right)} d t} = \ln{\left(\left|{\sin{\left(t \right)}}\right| \right)}$$

加上积分常数：

$$\int{\cot{\left(t \right)} d t} = \ln{\left(\left|{\sin{\left(t \right)}}\right| \right)}+C$$

$$$\int \cot{\left(t \right)}\, dt = \ln\left(\left|{\sin{\left(t \right)}}\right|\right) + C$$$A