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

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

改写被积函数:

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

对 $$$c=2$$$ 和 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{2 \cos{\left(x \right)} d x}}} = {\color{red}{\left(2 \int{\cos{\left(x \right)} d x}\right)}}$$

余弦函数的积分为 $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$：

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

因此，

$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}$$

加上积分常数：

$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}+C$$

$$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx = 2 \sin{\left(x \right)} + C$$$A