Integral de $$$x^{21} - x$$$

Encontre $$$\int \left(x^{21} - x\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(x^{21} - x\right)d x}}} = {\color{red}{\left(- \int{x d x} + \int{x^{21} d x}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=21$$$:

$$- \int{x d x} + {\color{red}{\int{x^{21} d x}}}=- \int{x d x} + {\color{red}{\frac{x^{1 + 21}}{1 + 21}}}=- \int{x d x} + {\color{red}{\left(\frac{x^{22}}{22}\right)}}$$

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

$$\frac{x^{22}}{22} - {\color{red}{\int{x d x}}}=\frac{x^{22}}{22} - {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\frac{x^{22}}{22} - {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

Portanto,

$$\int{\left(x^{21} - x\right)d x} = \frac{x^{22}}{22} - \frac{x^{2}}{2}$$

Simplifique:

$$\int{\left(x^{21} - x\right)d x} = \frac{x^{2} \left(x^{20} - 11\right)}{22}$$

Adicione a constante de integração:

$$\int{\left(x^{21} - x\right)d x} = \frac{x^{2} \left(x^{20} - 11\right)}{22}+C$$

$$$\int \left(x^{21} - x\right)\, dx = \frac{x^{2} \left(x^{20} - 11\right)}{22} + C$$$A