Integral de $$$_1 x^{3} - 1$$$ em relação a $$$x$$$

Encontre $$$\int \left(_1 x^{3} - 1\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(_1 x^{3} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} + \int{_1 x^{3} d x}\right)}}$$

Aplique a regra da constante $$$\int c\, dx = c x$$$ usando $$$c=1$$$:

$$\int{_1 x^{3} d x} - {\color{red}{\int{1 d x}}} = \int{_1 x^{3} d x} - {\color{red}{x}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=_1$$$ e $$$f{\left(x \right)} = x^{3}$$$:

$$- x + {\color{red}{\int{_1 x^{3} d x}}} = - x + {\color{red}{_1 \int{x^{3} d x}}}$$

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

$$_1 {\color{red}{\int{x^{3} d x}}} - x=_1 {\color{red}{\frac{x^{1 + 3}}{1 + 3}}} - x=_1 {\color{red}{\left(\frac{x^{4}}{4}\right)}} - x$$

Portanto,

$$\int{\left(_1 x^{3} - 1\right)d x} = \frac{_1 x^{4}}{4} - x$$

Adicione a constante de integração:

$$\int{\left(_1 x^{3} - 1\right)d x} = \frac{_1 x^{4}}{4} - x+C$$

$$$\int \left(_1 x^{3} - 1\right)\, dx = \left(\frac{_1 x^{4}}{4} - x\right) + C$$$A