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

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

Integre termo a termo:

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

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

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

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

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

Portanto,

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

Adicione a constante de integração:

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

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