Integral of $$$\frac{2 x}{3}$$$

Find $$$\int \frac{2 x}{3}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{2}{3}$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}{\int{\frac{2 x}{3} d x}}} = {\color{red}{\left(\frac{2 \int{x d x}}{3}\right)}}$$

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

$$\frac{2 {\color{red}{\int{x d x}}}}{3}=\frac{2 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}}{3}=\frac{2 {\color{red}{\left(\frac{x^{2}}{2}\right)}}}{3}$$

Therefore,

$$\int{\frac{2 x}{3} d x} = \frac{x^{2}}{3}$$

Add the constant of integration:

$$\int{\frac{2 x}{3} d x} = \frac{x^{2}}{3}+C$$

$$$\int \frac{2 x}{3}\, dx = \frac{x^{2}}{3} + C$$$A