Chapter 4.6, Problem 39E

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

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

Where E 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.

Find a formula for

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

$-e^{-kt}\left(\frac{T}{k}+\frac{1}{k^2}\right)+\frac{1}{k^2}$

Find

${\displaystyle {\int }_0^{10}E\left(t\right)dt,\text{ when }k=0.2mg/hr}$.