Integral de $$$- x + \frac{1}{a}$$$ em relação a $$$x$$$

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

Integre termo a termo:

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

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

$$- \int{x d x} + {\color{red}{\int{\frac{1}{a} d x}}} = - \int{x d x} + {\color{red}{\frac{x}{a}}}$$

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

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

Portanto,

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

Adicione a constante de integração:

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

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