$$$\frac{1}{\sqrt[6]{x}}$$$の積分

$$$\int \frac{1}{\sqrt[6]{x}}\, dx$$$ を求めよ。

$$$n=- \frac{1}{6}$$$ 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:

$${\color{red}{\int{\frac{1}{\sqrt[6]{x}} d x}}}={\color{red}{\int{x^{- \frac{1}{6}} d x}}}={\color{red}{\frac{x^{- \frac{1}{6} + 1}}{- \frac{1}{6} + 1}}}={\color{red}{\left(\frac{6 x^{\frac{5}{6}}}{5}\right)}}$$

したがって、

$$\int{\frac{1}{\sqrt[6]{x}} d x} = \frac{6 x^{\frac{5}{6}}}{5}$$

積分定数を加える:

$$\int{\frac{1}{\sqrt[6]{x}} d x} = \frac{6 x^{\frac{5}{6}}}{5}+C$$

$$$\int \frac{1}{\sqrt[6]{x}}\, dx = \frac{6 x^{\frac{5}{6}}}{5} + C$$$A