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

Find $$$\int \left(e^{x} + 2\right)\, dx$$$.

Integrate term by term:

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

Apply the constant rule $$$\int c\, dx = c x$$$ with $$$c=2$$$:

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

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

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

Therefore,

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

Add the constant of integration:

$$\int{\left(e^{x} + 2\right)d x} = 2 x + e^{x}+C$$

$$$\int \left(e^{x} + 2\right)\, dx = \left(2 x + e^{x}\right) + C$$$A