A particle moves in a straight line such that at time \(t\) seconds \((t \geqslant 0)\), its velocity \(v\), in \({\text{m}}{{\text{s}}^{ - 1}}\), is given by \(v = 10t{{\text{e}}^{ - 2t}}\). Find the exact distance travelled by the particle in the first half-second.

\(s = \int\limits_0^{\frac{1}{2}} {10t{{\text{e}}^{ - 2t}}{\text{d}}t} \)

attempt at integration by parts     M1

\( = \left[ { - 5t{{\text{e}}^{ - 2t}}} \right]_0^{\frac{1}{2}} - \int\limits_0^{\frac{1}{2}} { - 5{{\text{e}}^{ - 2t}}{\text{d}}t} \)     A1

\( = \left[ { - 5t{{\text{e}}^{ - 2t}} - \frac{5}{2}{{\text{e}}^{ - 2t}}} \right]_0^{\frac{1}{2}}\)     (A1)

Note:     Condone absence of limits (or incorrect limits) and missing factor of 10 up to this point.

\(s = \int\limits_0^{\frac{1}{2}} {10t{{\text{e}}^{ - 2t}}{\text{d}}t} \)     (M1)

\( =  - 5{{\text{e}}^{ - 1}} + \frac{5}{2}{\text{ }}\left( { = \frac{{ - 5}}{{\text{e}}} + \frac{5}{2}} \right){\text{ }}\left( { = \frac{{5{\text{e}} - 10}}{{2{\text{e}}}}} \right)\)     A1

[5 marks]