Integral de $$$\frac{x^{31}}{9}$$$

Encontre $$$\int \frac{x^{31}}{9}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=\frac{1}{9}$$$ e $$$f{\left(x \right)} = x^{31}$$$:

$${\color{red}{\int{\frac{x^{31}}{9} d x}}} = {\color{red}{\left(\frac{\int{x^{31} d x}}{9}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=31$$$:

$$\frac{{\color{red}{\int{x^{31} d x}}}}{9}=\frac{{\color{red}{\frac{x^{1 + 31}}{1 + 31}}}}{9}=\frac{{\color{red}{\left(\frac{x^{32}}{32}\right)}}}{9}$$

Portanto,

$$\int{\frac{x^{31}}{9} d x} = \frac{x^{32}}{288}$$

Adicione a constante de integração:

$$\int{\frac{x^{31}}{9} d x} = \frac{x^{32}}{288}+C$$

$$$\int \frac{x^{31}}{9}\, dx = \frac{x^{32}}{288} + C$$$A