Answer
The reaction rate to a new drug t hours after the drug is administered is
$$
r^{\prime}(t)=0.5t e^{-t}
$$
The total reaction over the first $5$ hours is given by:
$$
\begin{aligned}
0.5 \int_{0}^{5} t e^{-t} d t
&=\left.0.5\left(-t e^{-t}-e^{-t}\right)\right|_{0} ^{5} \\ &=0.5\left(-5 e^{-5}-e^{-5}+e^{0}\right) \\
& \approx 0.4798
\end{aligned}
$$
So, The total reaction over the first 5 hours is $ \approx 0.4798 $
Work Step by Step
The reaction rate to a new drug t hours after the drug is administered is
$$
r^{\prime}(t)=0.5t e^{-t}
$$
The total reaction over the first 5 hours is given by:
$$
\int_{0}^{\infty} 0.5t e^{-t} d t
$$
First, we evaluate the indefinite integral as follows:
$$
0.5\int t e^{-t} d t
$$
use integration by parts with
$$
\quad\quad\quad \left[\begin{array}{c}{u=t, \quad\quad dv= e^{-t} d t } \\ {d u= dt, \quad\quad v=\frac{e^{-t} }{-1} }\end{array}\right] ,
$$
$$
\begin{aligned}
\int t e^{-t} d t &=\frac{t e^{-t}}{-1}+\int e^{-t} d t \\
&=-t e^{-t}+\frac{e^{-t}}{-1} \\
\end{aligned}
$$
Now we will evaluate the improper integral as follows:
$$
\begin{aligned}
0.5 \int_{0}^{5} t e^{-t} d t
&=\left.0.5\left(-t e^{-t}-e^{-t}\right)\right|_{0} ^{5} \\ &=0.5\left(-5 e^{-5}-e^{-5}+e^{0}\right) \\
& \approx 0.4798
\end{aligned}
$$
So, The total reaction over the first 5 hours is $ \approx 0.4798 $
