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

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

Reescribe el integrando:

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

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=12$$$ y $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:

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

La integral del coseno es $$$\int{\cos{\left(x \right)} d x} = \sin{\left(x \right)}$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

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