A rocket moving in a straight line has velocity $v$ km s^–1 and displacement $s$ km at time $t$ seconds. The velocity $v$ is given by $v(t) = 6{{\rm{e}}^{2t}} + t$ . When $t = 0$ , $s = 10$ .

Find an expression for the displacement of the rocket in terms of $t$ .

evidence of anti-differentiation     (M1)

eg   $\int {(6{{\rm{e}}^{2t}} + t)}$

$s = 3{{\rm{e}}^{2t}} + \frac{{{t^2}}}{2} + C$     A2A1

Note: Award A2 for $3{{\rm{e}}^{2t}}$ , A1 for $\frac{{{t^2}}}{2}$ .

attempt to substitute ($0$, $10$) into their integrated expression (even if $C$ is missing)     (M1) 

correct working     (A1)

eg   $10 = 3 + C$ , $C = 7$

$s = 3{{\rm{e}}^{2t}} + \frac{{{t^2}}}{2} + 7$     A1     N6

Note: Exception to the FT rule. If working shown, allow full FT on incorrect integration which must involve a power of ${\rm{e}}$.

[7 marks]