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]