Integral de $$$\eta n - x^{3}$$$ em relação a $$$x$$$

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

Integre termo a termo:

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

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

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

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=\eta n$$$:

$$- \frac{x^{4}}{4} + {\color{red}{\int{\eta n d x}}} = - \frac{x^{4}}{4} + {\color{red}{\eta n x}}$$

Portanto,

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

Simplifique:

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

Adicione a constante de integração:

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

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