$$$\frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}}$$$ の $$$x$$$ に関する積分

$$$\int \frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}}\, dx$$$ を求めよ。

被積分関数を書き換える:

$${\color{red}{\int{\frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{4 \sin^{2}{\left(y \right)} \cos{\left(x \right)} \cos{\left(y \right)} d x}}}$$

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=4 \sin^{2}{\left(y \right)} \cos{\left(y \right)}$$$ と $$$f{\left(x \right)} = \cos{\left(x \right)}$$$ に対して適用する:

$${\color{red}{\int{4 \sin^{2}{\left(y \right)} \cos{\left(x \right)} \cos{\left(y \right)} d x}}} = {\color{red}{\left(4 \sin^{2}{\left(y \right)} \cos{\left(y \right)} \int{\cos{\left(x \right)} d x}\right)}}$$

余弦の積分は$$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

$$4 \sin^{2}{\left(y \right)} \cos{\left(y \right)} {\color{red}{\int{\cos{\left(x \right)} d x}}} = 4 \sin^{2}{\left(y \right)} \cos{\left(y \right)} {\color{red}{\sin{\left(x \right)}}}$$

したがって、

$$\int{\frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}} d x} = 4 \sin{\left(x \right)} \sin^{2}{\left(y \right)} \cos{\left(y \right)}$$

積分定数を加える:

$$\int{\frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}} d x} = 4 \sin{\left(x \right)} \sin^{2}{\left(y \right)} \cos{\left(y \right)}+C$$

$$$\int \frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}}\, dx = 4 \sin{\left(x \right)} \sin^{2}{\left(y \right)} \cos{\left(y \right)} + C$$$A