A patient takes a tablet, and the drug enters the blood from the stomach…

A patient takes a tablet, and the drug enters the blood from the stomach. At $t$ hours after the tablet is taken, the rate at which the drug is absorbed is $\dfrac{dA}{dt} = t^2e^{-t}$ milligrams per hour. No drug has been absorbed at $t = 0$, so $A(0) = 0\ \text{mg}$. The doctor wants the total amount absorbed by the time $t$. The integral needs integration by parts twice, because $t^2$ must be reduced to a constant. She takes $u = t^2$ first and then $u = t$. After her working she writes $A(t) = K - e^{-t}\left(t^2 + 2t + 2\right)$. Find the constant $K$. Then find the amount absorbed after $1$ hour and after $3$ hours. The doctor also asks what total amount the body can ever absorb. As $t$ grows very large, the term with $e^{-t}$ tends to zero. Give $K$ in milligrams, and each amount in milligrams to four decimal places.