$$$- \frac{\sin{\left(x \right)}}{\sin{\left(a - x \right)}}$$$ 對 $$$x$$$ 的積分

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

重寫被積函數:

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

重寫分子並將分式拆分:

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

逐項積分：

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

配合 $$$c=\frac{\cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$$，應用常數法則 $$$\int c\, dx = c x$$$：

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

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

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

該積分可改寫為

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=\frac{\sin{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$$ 與 $$$f{\left(u \right)} = \frac{1}{u}$$$：

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

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

$$\frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\sin{\left(a \right)} {\color{red}{\int{\frac{1}{u} d u}}}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} = \frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\sin{\left(a \right)} {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$

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

$$\frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)} \sin{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} = \frac{x \cos{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}} + \frac{\ln{\left(\left|{{\color{red}{\left(\sin{\left(a \right)} \cos{\left(x \right)} - \sin{\left(x \right)} \cos{\left(a \right)}\right)}}}\right| \right)} \sin{\left(a \right)}}{\sin^{2}{\left(a \right)} + \cos^{2}{\left(a \right)}}$$

因此，

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

化簡：

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

加上積分常數：

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

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