Integral of $$$\frac{1}{\sqrt[6]{x}}$$$

Find $$$\int \frac{1}{\sqrt[6]{x}}\, dx$$$.

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=- \frac{1}{6}$$$:

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

Therefore,

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

Add the constant of integration:

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

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