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

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

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

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

Logo,

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

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

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

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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