Integral de $$$- x^{2} + \frac{1}{a^{2}}$$$ em relação a $$$x$$$

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

Integre termo a termo:

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

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

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

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

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

Portanto,

$$\int{\left(- x^{2} + \frac{1}{a^{2}}\right)d x} = - \frac{x^{3}}{3} + \frac{x}{a^{2}}$$

Adicione a constante de integração:

$$\int{\left(- x^{2} + \frac{1}{a^{2}}\right)d x} = - \frac{x^{3}}{3} + \frac{x}{a^{2}}+C$$

$$$\int \left(- x^{2} + \frac{1}{a^{2}}\right)\, dx = \left(- \frac{x^{3}}{3} + \frac{x}{a^{2}}\right) + C$$$A