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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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