Integral de $$$x^{4} - x^{3}$$$

Encontre $$$\int \left(x^{4} - x^{3}\right)\, dx$$$.

Integre termo a termo:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=4$$$:

$$- \int{x^{3} d x} + {\color{red}{\int{x^{4} d x}}}=- \int{x^{3} d x} + {\color{red}{\frac{x^{1 + 4}}{1 + 4}}}=- \int{x^{3} d x} + {\color{red}{\left(\frac{x^{5}}{5}\right)}}$$

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

$$\frac{x^{5}}{5} - {\color{red}{\int{x^{3} d x}}}=\frac{x^{5}}{5} - {\color{red}{\frac{x^{1 + 3}}{1 + 3}}}=\frac{x^{5}}{5} - {\color{red}{\left(\frac{x^{4}}{4}\right)}}$$

Portanto,

$$\int{\left(x^{4} - x^{3}\right)d x} = \frac{x^{5}}{5} - \frac{x^{4}}{4}$$

Simplifique:

$$\int{\left(x^{4} - x^{3}\right)d x} = \frac{x^{4} \left(4 x - 5\right)}{20}$$

Adicione a constante de integração:

$$\int{\left(x^{4} - x^{3}\right)d x} = \frac{x^{4} \left(4 x - 5\right)}{20}+C$$

$$$\int \left(x^{4} - x^{3}\right)\, dx = \frac{x^{4} \left(4 x - 5\right)}{20} + C$$$A