Integral of $$$3 e^{t}$$$

Find $$$\int 3 e^{t}\, dt$$$.

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

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

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

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

Therefore,

$$\int{3 e^{t} d t} = 3 e^{t}$$

Add the constant of integration:

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

$$$\int 3 e^{t}\, dt = 3 e^{t} + C$$$A