$$$x$$$에 대한 $$$\frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}}$$$의 적분

$$$\int \frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}}\, dx$$$을(를) 구하시오.

피적분함수를 다시 쓰십시오:

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

상수배 법칙 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$을 $$$c=4 \sin^{2}{\left(y \right)} \cos{\left(y \right)}$$$와 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$에 적용하세요:

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

코사인의 적분은 $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

따라서,

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

적분 상수를 추가하세요:

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

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