WAEC Further Mathematics 2012 Paper

Concept:Use the power rule of integration to find the antiderivative, then apply the limits of the definite integral using $F(b) - F(a)$.Explanation:We need to evaluate $\int_{-2}^{3} (3x^{2} - 2x - 12) \, dx$.First, integrate each term using $\int x^n \, dx = \frac{x^{n+1}}{n+1}$.$\int 3x^{2} \, dx = 3 \cdot \frac{x^{3}}{3} = x^{3}$.$\int (-2x) \, dx = -2 \cdot \frac{x^{2}}{2} = -x^{2}$.$\int (-12) \, dx = -12x$.So the antiderivative is $F(x) = x^{3} - x^{2} - 12x$.Now substitute the upper and lower limits:$F(3) = 3^{3} - 3^{2} - 12(3) = 27 - 9 - 36 = -18$.$F(-2) = (-2)^{3} - (-2)^{2} - 12(-2) = -8 - 4 + 24 = 12$.Therefore, the definite integral is $F(3) - F(-2) = -18 - 12 = -30$.Answer:The value of the integral is $-30$, so the correct option is A ($-30$).