$$$\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}$$$ 的積分
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求$$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx$$$。
解答
重寫被積函數:
$${\color{red}{\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{2 \cos{\left(x \right)} d x}}}$$
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=2$$$ 與 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$：
$${\color{red}{\int{2 \cos{\left(x \right)} d x}}} = {\color{red}{\left(2 \int{\cos{\left(x \right)} d x}\right)}}$$
餘弦函數的積分為 $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$：
$$2 {\color{red}{\int{\cos{\left(x \right)} d x}}} = 2 {\color{red}{\sin{\left(x \right)}}}$$
因此，
$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}$$
加上積分常數：
$$\int{\frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 2 \sin{\left(x \right)}+C$$
答案
$$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx = 2 \sin{\left(x \right)} + C$$$A