Integral de $$$\frac{1}{2} - x$$$

Encontre $$$\int \left(\frac{1}{2} - x\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(\frac{1}{2} - x\right)d x}}} = {\color{red}{\left(\int{\frac{1}{2} d x} - \int{x d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=\frac{1}{2}$$$:

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

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

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

Portanto,

$$\int{\left(\frac{1}{2} - x\right)d x} = - \frac{x^{2}}{2} + \frac{x}{2}$$

Simplifique:

$$\int{\left(\frac{1}{2} - x\right)d x} = \frac{x \left(1 - x\right)}{2}$$

Adicione a constante de integração:

$$\int{\left(\frac{1}{2} - x\right)d x} = \frac{x \left(1 - x\right)}{2}+C$$

$$$\int \left(\frac{1}{2} - x\right)\, dx = \frac{x \left(1 - x\right)}{2} + C$$$A