Integral de $$$\frac{10}{x^{\frac{7}{5}}}$$$

Encontre $$$\int \frac{10}{x^{\frac{7}{5}}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=10$$$ e $$$f{\left(x \right)} = \frac{1}{x^{\frac{7}{5}}}$$$:

$${\color{red}{\int{\frac{10}{x^{\frac{7}{5}}} d x}}} = {\color{red}{\left(10 \int{\frac{1}{x^{\frac{7}{5}}} d x}\right)}}$$

Aplique a regra da potência $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=- \frac{7}{5}$$$:

$$10 {\color{red}{\int{\frac{1}{x^{\frac{7}{5}}} d x}}}=10 {\color{red}{\int{x^{- \frac{7}{5}} d x}}}=10 {\color{red}{\frac{x^{- \frac{7}{5} + 1}}{- \frac{7}{5} + 1}}}=10 {\color{red}{\left(- \frac{5 x^{- \frac{2}{5}}}{2}\right)}}=10 {\color{red}{\left(- \frac{5}{2 x^{\frac{2}{5}}}\right)}}$$

Portanto,

$$\int{\frac{10}{x^{\frac{7}{5}}} d x} = - \frac{25}{x^{\frac{2}{5}}}$$

Adicione a constante de integração:

$$\int{\frac{10}{x^{\frac{7}{5}}} d x} = - \frac{25}{x^{\frac{2}{5}}}+C$$

$$$\int \frac{10}{x^{\frac{7}{5}}}\, dx = - \frac{25}{x^{\frac{2}{5}}} + C$$$A