Integral of $$$\frac{1}{x^{\frac{5}{6}}}$$$

Find $$$\int \frac{1}{x^{\frac{5}{6}}}\, dx$$$.

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=- \frac{5}{6}$$$:

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

Therefore,

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

Add the constant of integration:

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

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