$$$\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$$ 的積分

求$$$\int \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\, dx$$$。

令 $$$u=1 - \cos{\left(x \right)}$$$。

則 $$$du=\left(1 - \cos{\left(x \right)}\right)^{\prime }dx = \sin{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\sin{\left(x \right)} dx = du$$$。

因此，

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

$$$\frac{1}{u}$$$ 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 $$$u=1 - \cos{\left(x \right)}$$$：

$$\ln{\left(\left|{{\color{red}{u}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\left(1 - \cos{\left(x \right)}\right)}}}\right| \right)}$$

因此，

$$\int{\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}} d x} = \ln{\left(\left|{\cos{\left(x \right)} - 1}\right| \right)}$$

加上積分常數：

$$\int{\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}} d x} = \ln{\left(\left|{\cos{\left(x \right)} - 1}\right| \right)}+C$$

$$$\int \frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\, dx = \ln\left(\left|{\cos{\left(x \right)} - 1}\right|\right) + C$$$A