Integral de $$$\frac{\cos{\left(4 t \right)}}{2}$$$

Halla $$$\int \frac{\cos{\left(4 t \right)}}{2}\, dt$$$.

Aplica la regla del factor constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ con $$$c=\frac{1}{2}$$$ y $$$f{\left(t \right)} = \cos{\left(4 t \right)}$$$:

$${\color{red}{\int{\frac{\cos{\left(4 t \right)}}{2} d t}}} = {\color{red}{\left(\frac{\int{\cos{\left(4 t \right)} d t}}{2}\right)}}$$

Sea $$$u=4 t$$$.

Entonces $$$du=\left(4 t\right)^{\prime }dt = 4 dt$$$ (los pasos pueden verse »), y obtenemos que $$$dt = \frac{du}{4}$$$.

Entonces,

$$\frac{{\color{red}{\int{\cos{\left(4 t \right)} d t}}}}{2} = \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2}$$

Aplica la regla del factor constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ con $$$c=\frac{1}{4}$$$ y $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

$$\frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{4} d u}}}}{2} = \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{4}\right)}}}{2}$$

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

$$\frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{8} = \frac{{\color{red}{\sin{\left(u \right)}}}}{8}$$

Recordemos que $$$u=4 t$$$:

$$\frac{\sin{\left({\color{red}{u}} \right)}}{8} = \frac{\sin{\left({\color{red}{\left(4 t\right)}} \right)}}{8}$$

Por lo tanto,

$$\int{\frac{\cos{\left(4 t \right)}}{2} d t} = \frac{\sin{\left(4 t \right)}}{8}$$

Añade la constante de integración:

$$\int{\frac{\cos{\left(4 t \right)}}{2} d t} = \frac{\sin{\left(4 t \right)}}{8}+C$$

$$$\int \frac{\cos{\left(4 t \right)}}{2}\, dt = \frac{\sin{\left(4 t \right)}}{8} + C$$$A