Integral de $$$\frac{5}{x^{5}}$$$

Encontre $$$\int \frac{5}{x^{5}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=5$$$ e $$$f{\left(x \right)} = \frac{1}{x^{5}}$$$:

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

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=-5$$$:

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

Portanto,

$$\int{\frac{5}{x^{5}} d x} = - \frac{5}{4 x^{4}}$$

Adicione a constante de integração:

$$\int{\frac{5}{x^{5}} d x} = - \frac{5}{4 x^{4}}+C$$

$$$\int \frac{5}{x^{5}}\, dx = - \frac{5}{4 x^{4}} + C$$$A