Let $f'(x) = 6{x^2} - 5$. Given that $f(2) =  - 3$, find $f(x)$.

evidence of antidifferentiation     (M1)

eg$\;\;\;f = \int {f'}$

correct integration (accept absence of $C$)     (A1)(A1)

$f(x) = \frac{{6{x^3}}}{3} - 5x + C,{\text{ }}2{x^3} - 5x$

attempt to substitute $(2,{\text{ }} - 3)$ into their integrated expression (must have $C$)     M1

eg$\;\;\;2{(2)^3} - 5(2) + C =  - 3,{\text{ }}16 - 10 + C =  - 3$

Note:     Award M0 if substituted into original or differentiated function.

correct working to find $C$     (A1)

eg$\;\;\;16 - 10 + C =  - 3,{\text{ }}6 + C =  - 3,{\text{ }}C =  - 9$

$f(x) = 2{x^3} - 5x - 9$     A1     N4

[6 marks]