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]