NECO 2023 · Paper 1 · Q17

A curve passes through $(-2, 3)$ and its gradient at any point is $3x^2 + 8x - 5$ . Find the equation of the curve.

Worked solution (try it first)

Integrate the gradient term by term: $y = x^3 + 4x^2 - 5x + c$ .
The curve passes through $(-2, 3)$ , so put $x = -2$ and $y = 3$ : $3 = -8 + 16 + 10 + c$ , that is $3 = 18 + c$ .
So $c = -15$ and $y = x^3 + 4x^2 - 5x - 15$ , option D.