Integral of $$$\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}}$$$

Find $$$\int \frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx$$$.

Rewrite the integrand:

$${\color{red}{\int{\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x}}} = {\color{red}{\int{12 \cos{\left(x \right)} d x}}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=12$$$ and $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

$${\color{red}{\int{12 \cos{\left(x \right)} d x}}} = {\color{red}{\left(12 \int{\cos{\left(x \right)} d x}\right)}}$$

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

$$12 {\color{red}{\int{\cos{\left(x \right)} d x}}} = 12 {\color{red}{\sin{\left(x \right)}}}$$

Therefore,

$$\int{\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 12 \sin{\left(x \right)}$$

Add the constant of integration:

$$\int{\frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}} d x} = 12 \sin{\left(x \right)}+C$$

$$$\int \frac{6 \sin{\left(2 x \right)}}{\sin{\left(x \right)}}\, dx = 12 \sin{\left(x \right)} + C$$$A