Integral of $$$5 e^{t}$$$

Find $$$\int 5 e^{t}\, dt$$$.

Apply the constant multiple rule $$$\int c f{\left(t \right)}\, dt = c \int f{\left(t \right)}\, dt$$$ with $$$c=5$$$ and $$$f{\left(t \right)} = e^{t}$$$:

$${\color{red}{\int{5 e^{t} d t}}} = {\color{red}{\left(5 \int{e^{t} d t}\right)}}$$

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

$$5 {\color{red}{\int{e^{t} d t}}} = 5 {\color{red}{e^{t}}}$$

Therefore,

$$\int{5 e^{t} d t} = 5 e^{t}$$

Add the constant of integration:

$$\int{5 e^{t} d t} = 5 e^{t}+C$$

$$$\int 5 e^{t}\, dt = 5 e^{t} + C$$$A