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

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

Aplique a regra do múltiplo constante $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ usando $$$c=\frac{1}{2}$$$ e $$$f{\left(t \right)} = \cos{\left(4 t \right)}$$$:

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

Seja $$$u=4 t$$$.

Então $$$du=\left(4 t\right)^{\prime }dt = 4 dt$$$ (veja os passos »), e obtemos $$$dt = \frac{du}{4}$$$.

Logo,

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=\frac{1}{4}$$$ e $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

A integral do cosseno é $$$\int{\cos{\left(u \right)} d u} = \sin{\left(u \right)}$$$:

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

Recorde que $$$u=4 t$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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