Integral de $$$\sin{\left(2 \theta \right)}$$$

Halla $$$\int \sin{\left(2 \theta \right)}\, d\theta$$$.

Sea $$$u=2 \theta$$$.

Entonces $$$du=\left(2 \theta\right)^{\prime }d\theta = 2 d\theta$$$ (los pasos pueden verse »), y obtenemos que $$$d\theta = \frac{du}{2}$$$.

La integral se convierte en

$${\color{red}{\int{\sin{\left(2 \theta \right)} d \theta}}} = {\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$${\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}} = {\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{2}\right)}}$$

La integral del seno es $$$\int{\sin{\left(u \right)} d u} = - \cos{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{2} = \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{2}$$

Recordemos que $$$u=2 \theta$$$:

$$- \frac{\cos{\left({\color{red}{u}} \right)}}{2} = - \frac{\cos{\left({\color{red}{\left(2 \theta\right)}} \right)}}{2}$$

Por lo tanto,

$$\int{\sin{\left(2 \theta \right)} d \theta} = - \frac{\cos{\left(2 \theta \right)}}{2}$$

Añade la constante de integración:

$$\int{\sin{\left(2 \theta \right)} d \theta} = - \frac{\cos{\left(2 \theta \right)}}{2}+C$$

$$$\int \sin{\left(2 \theta \right)}\, d\theta = - \frac{\cos{\left(2 \theta \right)}}{2} + C$$$A