Answer
$$ - 37{e^{ - 5}} + 2$$
Work Step by Step
$$\eqalign{
& {\text{The travel of the particle is given by}} \cr
& s\left( t \right) = \int_0^5 {{t^2}{e^{ - t}}dt} \cr
& {\text{Integrate by tabulation, see image below}} \cr
& s\left( t \right) = \left[ {{t^2}\left( { - {e^{ - t}}} \right) + 2t\left( { - {e^{ - t}}} \right) + 2\left( { - {e^{ - t}}} \right)} \right]_0^5 \cr
& s\left( t \right) = \left[ { - \left( {{t^2} + 2t + 2} \right){e^{ - t}}} \right]_0^5 \cr
& {\text{Evaluating}} \cr
& = \left[ { - \left( {{5^2} + 10 + 2} \right){e^{ - 5}}} \right] - \left[ { - \left( {{0^2} + 0 + 2} \right){e^{ - 0}}} \right] \cr
& = - 37{e^{ - 5}} + 2{e^0} \cr
& = - 37{e^{ - 5}} + 2 \cr} $$
