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

Halla $$$\int 2 \sin{\left(2 t \right)}\, dt$$$.

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

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

Sea $$$u=2 t$$$.

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

La integral se convierte en

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

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 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

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

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

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

Recordemos que $$$u=2 t$$$:

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

Por lo tanto,

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

Añade la constante de integración:

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

$$$\int 2 \sin{\left(2 t \right)}\, dt = - \cos{\left(2 t \right)} + C$$$A