Integral de $$$x^{2} - \frac{12}{x^{31}}$$$

Encontre $$$\int \left(x^{2} - \frac{12}{x^{31}}\right)\, dx$$$.

Integre termo a termo:

$${\color{red}{\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x}}} = {\color{red}{\left(- \int{\frac{12}{x^{31}} d x} + \int{x^{2} d x}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

$$- \int{\frac{12}{x^{31}} d x} + {\color{red}{\int{x^{2} d x}}}=- \int{\frac{12}{x^{31}} d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- \int{\frac{12}{x^{31}} d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=12$$$ e $$$f{\left(x \right)} = \frac{1}{x^{31}}$$$:

$$\frac{x^{3}}{3} - {\color{red}{\int{\frac{12}{x^{31}} d x}}} = \frac{x^{3}}{3} - {\color{red}{\left(12 \int{\frac{1}{x^{31}} d x}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=-31$$$:

$$\frac{x^{3}}{3} - 12 {\color{red}{\int{\frac{1}{x^{31}} d x}}}=\frac{x^{3}}{3} - 12 {\color{red}{\int{x^{-31} d x}}}=\frac{x^{3}}{3} - 12 {\color{red}{\frac{x^{-31 + 1}}{-31 + 1}}}=\frac{x^{3}}{3} - 12 {\color{red}{\left(- \frac{x^{-30}}{30}\right)}}=\frac{x^{3}}{3} - 12 {\color{red}{\left(- \frac{1}{30 x^{30}}\right)}}$$

Portanto,

$$\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x} = \frac{x^{3}}{3} + \frac{2}{5 x^{30}}$$

Simplifique:

$$\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x} = \frac{5 x^{33} + 6}{15 x^{30}}$$

Adicione a constante de integração:

$$\int{\left(x^{2} - \frac{12}{x^{31}}\right)d x} = \frac{5 x^{33} + 6}{15 x^{30}}+C$$

$$$\int \left(x^{2} - \frac{12}{x^{31}}\right)\, dx = \frac{5 x^{33} + 6}{15 x^{30}} + C$$$A