$$$\frac{10}{x^{\frac{7}{5}}}$$$の積分

$$$\int \frac{10}{x^{\frac{7}{5}}}\, dx$$$ を求めよ。

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=10$$$ と $$$f{\left(x \right)} = \frac{1}{x^{\frac{7}{5}}}$$$ に対して適用する:

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

$$$n=- \frac{7}{5}$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

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

したがって、

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

積分定数を加える:

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

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