Chapter 5.3, Problem 53E

Suppose an oral dose of a drug is taken. Over, time, the drug is assimilated in the body and excreted though the urine. The total amount of the drug that has passed through the body in T hours is given by

${\displaystyle {\int }_0^{T}E\left(t\right)}dt,$

where $E\left(t\right)$

is the rate of excretion of the drug. A typical rate-of-excretion function is $E\left(t\right)=t{e}^{-kt},$ where $k>0$ and t is the time, in hours. Use this information for Exercises 49-50.

Find ${\displaystyle {\int }_0^{\infty }E\left(t\right)}dt,$

and interpret the answer. That is, what does the integral represent?