Integral de $$$- \frac{8 x^{2}}{9}$$$

Encontre $$$\int \left(- \frac{8 x^{2}}{9}\right)\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=- \frac{8}{9}$$$ e $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}{\int{\left(- \frac{8 x^{2}}{9}\right)d x}}} = {\color{red}{\left(- \frac{8 \int{x^{2} d x}}{9}\right)}}$$

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

$$- \frac{8 {\color{red}{\int{x^{2} d x}}}}{9}=- \frac{8 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}}{9}=- \frac{8 {\color{red}{\left(\frac{x^{3}}{3}\right)}}}{9}$$

Portanto,

$$\int{\left(- \frac{8 x^{2}}{9}\right)d x} = - \frac{8 x^{3}}{27}$$

Adicione a constante de integração:

$$\int{\left(- \frac{8 x^{2}}{9}\right)d x} = - \frac{8 x^{3}}{27}+C$$

$$$\int \left(- \frac{8 x^{2}}{9}\right)\, dx = - \frac{8 x^{3}}{27} + C$$$A