Let \(f'(x) = 3{x^2} + 2\) . Given that \(f(2) = 5\) , find \(f(x)\) .

evidence of anti-differentiation     (M1)

e.g.  \(\int {f'(x)} \) , \(\int {(3{x^2} + 2){\rm{d}}x} \)

\(f(x) = {x^3} + 2x + c\) (seen anywhere, including the answer)     A1A1

attempt to substitute (2, 5)     (M1)

e.g. \(f(2) = {(2)^3} + 2(2)\) , \(5 = 8 + 4 + c\)

finding the value of c     (A1)

e.g. \(5 = 12 + c\) , \(c = - 7\)

\(f(x) = {x^3} + 2x - 7\)     A1     N5

[6 marks]

This question, which required candidates to integrate a simple polynomial and then substitute an initial condition to solve for "c", was very well done. Nearly all candidates who attempted this question were able to earn full marks. The very few mistakes that were seen involved arithmetic errors when solving for "c", or failing to write the final answer as the equation of the function.