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

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

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 $$$f{\left(t \right)} = \cos{\left(\frac{t}{2} \right)}$$$:

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

Seja $$$u=\frac{t}{2}$$$.

Então $$$du=\left(\frac{t}{2}\right)^{\prime }dt = \frac{dt}{2}$$$ (veja os passos »), e obtemos $$$dt = 2 du$$$.

A integral pode ser reescrita como

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

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=2$$$ e $$$f{\left(u \right)} = \cos{\left(u \right)}$$$:

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

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

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

Recorde que $$$u=\frac{t}{2}$$$:

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

Portanto,

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

Adicione a constante de integração:

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

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