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

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

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

$${\color{red}{\int{\sqrt[6]{x} d x}}}={\color{red}{\int{x^{\frac{1}{6}} d x}}}={\color{red}{\frac{x^{\frac{1}{6} + 1}}{\frac{1}{6} + 1}}}={\color{red}{\left(\frac{6 x^{\frac{7}{6}}}{7}\right)}}$$

Therefore,

$$\int{\sqrt[6]{x} d x} = \frac{6 x^{\frac{7}{6}}}{7}$$

Add the constant of integration:

$$\int{\sqrt[6]{x} d x} = \frac{6 x^{\frac{7}{6}}}{7}+C$$

$$$\int \sqrt[6]{x}\, dx = \frac{6 x^{\frac{7}{6}}}{7} + C$$$A