Drug Reaction The reaction rate to a new drug t hours after the drug is administered is

r' (t) = 0.5te^-t.

Find the total reaction over the first 5 hours.

0.4865

Step-by-step explanation:

The reaction rate to a new drug t hours after the drug is administered is given by the expression r' (t) = 0.5te^-t.

First, we find an expression for r (t), which is the total reaction to the drug, stating from t=0, when the drug was administered by integrating r' (t).

Here, we employ the "{" sign to represent integration, and apply the method of integrating by parts as follows: {udv = uv - {vdu

r = {0.5te^-tdt

To do this we let u = 0.5t, and dv = (e^-t) dt

Thus du = 0.5 dt and v = - e^-t

Therefore r = 0.5 t * ( - e^-t) - {0.5 * - e^-t dt =

= - 0.5 t * (e^-t) - 0.5 * e^-t + C

To evaluate the C, we plug in r = 0 at t=0

0 = 0-0.5+C

c = 0.5

Thus r (t) = 0.5 - 0.5 t * (e^-t) - 0.5 * e^-t = 0.5 - (0.5e^-t) (t - 1)

Over the first 5 hours we obtain r (5) as follows:

r (5) = 0.5 - 0.5*5 * (e^-5) - 0.5 * e^--5 = 0.4865