$$$\frac{2 \sin{\left(x \right)}}{\sin{\left(x \right)} - \cos{\left(x \right)}}$$$ 的积分

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

改写分子并拆分分式:

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

逐项积分：

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

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

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

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

则$$$du=\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{\prime }dx = \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) dx$$$ (步骤见»)，并有$$$\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) dx = du$$$。

该积分可以改写为

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

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

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

回忆一下 $$$u=\sin{\left(x \right)} - \cos{\left(x \right)}$$$:

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

因此，

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

化简：

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

加上积分常数（并从表达式中去除常数项）：

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

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