Integral of $$$\frac{\sin{\left(2 x \right)} \sin{\left(y \right)} \sin{\left(2 y \right)}}{\sin{\left(x \right)}}$$$ with respect to $$$x$$$

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

Rewrite the integrand:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4 \sin^{2}{\left(y \right)} \cos{\left(y \right)}$$$ and $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

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

The integral of the cosine is $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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