Integral of $$$d^{2} e^{t}$$$ with respect to $$$t$$$

Find $$$\int d^{2} e^{t}\, dt$$$.

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

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

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

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

Therefore,

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

Add the constant of integration:

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

$$$\int d^{2} e^{t}\, dt = d^{2} e^{t} + C$$$A