Integral of $$$\frac{e^{x}}{3}$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{3}$$$ and $$$f{\left(x \right)} = e^{x}$$$:

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

The integral of the exponential function is $$$\int{e^{x} d x} = e^{x}$$$:

$$\frac{{\color{red}{\int{e^{x} d x}}}}{3} = \frac{{\color{red}{e^{x}}}}{3}$$

Therefore,

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

Add the constant of integration:

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

Answer: $$$\int{\frac{e^{x}}{3} d x}=\frac{e^{x}}{3}+C$$$