Integral de $$$8 \sqrt[3]{x}$$$

Encontre $$$\int 8 \sqrt[3]{x}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=8$$$ e $$$f{\left(x \right)} = \sqrt[3]{x}$$$:

$${\color{red}{\int{8 \sqrt[3]{x} d x}}} = {\color{red}{\left(8 \int{\sqrt[3]{x} d x}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=\frac{1}{3}$$$:

$$8 {\color{red}{\int{\sqrt[3]{x} d x}}}=8 {\color{red}{\int{x^{\frac{1}{3}} d x}}}=8 {\color{red}{\frac{x^{\frac{1}{3} + 1}}{\frac{1}{3} + 1}}}=8 {\color{red}{\left(\frac{3 x^{\frac{4}{3}}}{4}\right)}}$$

Portanto,

$$\int{8 \sqrt[3]{x} d x} = 6 x^{\frac{4}{3}}$$

Adicione a constante de integração:

$$\int{8 \sqrt[3]{x} d x} = 6 x^{\frac{4}{3}}+C$$

$$$\int 8 \sqrt[3]{x}\, dx = 6 x^{\frac{4}{3}} + C$$$A