Integral de $$$- x^{3}$$$

Encontre $$$\int \left(- x^{3}\right)\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=-1$$$ e $$$f{\left(x \right)} = x^{3}$$$:

$${\color{red}{\int{\left(- x^{3}\right)d x}}} = {\color{red}{\left(- \int{x^{3} d x}\right)}}$$

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

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

Portanto,

$$\int{\left(- x^{3}\right)d x} = - \frac{x^{4}}{4}$$

Adicione a constante de integração:

$$\int{\left(- x^{3}\right)d x} = - \frac{x^{4}}{4}+C$$

$$$\int \left(- x^{3}\right)\, dx = - \frac{x^{4}}{4} + C$$$A