Integral of $$$\frac{e^{x}}{2}$$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(x \right)} = e^{x}$$$:

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

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

$$\frac{{\color{red}{\int{e^{x} d x}}}}{2} = \frac{{\color{red}{e^{x}}}}{2}$$

Therefore,

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

Add the constant of integration:

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

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