$$$\cot{\left(v \right)}$$$ 的積分

求$$$\int \cot{\left(v \right)}\, dv$$$。

將餘切改寫為 $$$\cot\left(v\right)=\frac{\cos\left(v\right)}{\sin\left(v\right)}$$$:

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

令 $$$u=\sin{\left(v \right)}$$$。

則 $$$du=\left(\sin{\left(v \right)}\right)^{\prime }dv = \cos{\left(v \right)} dv$$$ (步驟見»)，並可得 $$$\cos{\left(v \right)} dv = du$$$。

因此，

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

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

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

回顧一下 $$$u=\sin{\left(v \right)}$$$：

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

因此，

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

加上積分常數：

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

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