$$$\frac{1}{x^{\frac{7}{5}}}$$$の積分

$$$\int \frac{1}{x^{\frac{7}{5}}}\, dx$$$ を求めよ。

$$$n=- \frac{7}{5}$$$ 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:

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

したがって、

$$\int{\frac{1}{x^{\frac{7}{5}}} d x} = - \frac{5}{2 x^{\frac{2}{5}}}$$

積分定数を加える:

$$\int{\frac{1}{x^{\frac{7}{5}}} d x} = - \frac{5}{2 x^{\frac{2}{5}}}+C$$

$$$\int \frac{1}{x^{\frac{7}{5}}}\, dx = - \frac{5}{2 x^{\frac{2}{5}}} + C$$$A