$$$\frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}$$$ 的積分

求$$$\int \frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}\, dx$$$。

重寫分子並將分式拆分:

$${\color{red}{\int{\frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} d x}}} = {\color{red}{\int{\left(1 + \frac{\sin{\left(x \right)} + \cos{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}\right)d x}}}$$

逐項積分：

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

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

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

令 $$$u=\sin{\left(x \right)} - \cos{\left(x \right)}$$$。

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

因此，

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

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

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

回顧一下 $$$u=\sin{\left(x \right)} - \cos{\left(x \right)}$$$：

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

因此，

$$\int{\frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} d x} = x + \ln{\left(\left|{\sin{\left(x \right)} - \cos{\left(x \right)}}\right| \right)}$$

化簡：

$$\int{\frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} d x} = x + \ln{\left(\left|{\cos{\left(x + \frac{\pi}{4} \right)}}\right| \right)} + \frac{\ln{\left(2 \right)}}{2}$$

加上積分常數（並從表達式中移除常數項）：

$$\int{\frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}} d x} = x + \ln{\left(\left|{\cos{\left(x + \frac{\pi}{4} \right)}}\right| \right)}+C$$

$$$\int \frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}\, dx = \left(x + \ln\left(\left|{\cos{\left(x + \frac{\pi}{4} \right)}}\right|\right)\right) + C$$$A