Integral de $$$\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}$$$

Encontre $$$\int \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}\, dt$$$.

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 $$$f{\left(t \right)} = \left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}$$$:

$${\color{red}{\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t}}} = {\color{red}{\left(\frac{\int{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)} d t}}{2}\right)}}$$

Seja $$$u=1 - \frac{\sin{\left(t \right)}}{2}$$$.

Então $$$du=\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{\prime }dt = - \frac{\cos{\left(t \right)}}{2} dt$$$ (veja os passos »), e obtemos $$$\cos{\left(t \right)} dt = - 2 du$$$.

A integral pode ser reescrita como

$$\frac{{\color{red}{\int{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)} d t}}}}{2} = \frac{{\color{red}{\int{\left(- 2 u^{2}\right)d u}}}}{2}$$

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

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

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

$$- {\color{red}{\int{u^{2} d u}}}=- {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=- {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

Recorde que $$$u=1 - \frac{\sin{\left(t \right)}}{2}$$$:

$$- \frac{{\color{red}{u}}^{3}}{3} = - \frac{{\color{red}{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)}}^{3}}{3}$$

Portanto,

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = - \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{3}}{3}$$

Simplifique:

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24}$$

Adicione a constante de integração:

$$\int{\frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2} d t} = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24}+C$$

$$$\int \frac{\left(1 - \frac{\sin{\left(t \right)}}{2}\right)^{2} \cos{\left(t \right)}}{2}\, dt = \frac{\left(\sin{\left(t \right)} - 2\right)^{3}}{24} + C$$$A