WAEC Further Mathematics 2018 Paper

Concept:Simplify the integrand into separate terms before integrating using the power rule.Explanation:Rewrite the fraction term by term:$\int_{1/2}^{1} \frac{x^3 - 4}{x^3} \, dx = \int_{1/2}^{1} \left(1 - \frac{4}{x^3}\right) dx$Express the second term using a negative exponent:$\int_{1/2}^{1} \left(1 - 4x^{-3}\right) dx$Integrate each term separately:$\int 1 \, dx = x, \quad \int -4x^{-3} \, dx = -4 \cdot \frac{x^{-2}}{-2} = \frac{2}{x^2}$So the antiderivative is:$x + \frac{2}{x^2}$Apply the limits from $x = \frac{1}{2}$ to $x = 1$:At $x = 1$:$1 + \frac{2}{1^2} = 3$At $x = \frac{1}{2}$:$\frac{1}{2} + \frac{2}{\left(\frac{1}{2}\right)^2} = \frac{1}{2} + \frac{2}{\frac{1}{4}} = 0.5 + 8 = 8.5$Subtract the lower limit from the upper limit:$3 - 8.5 = -5.5$Answer:The value of the definite integral is $-5.5$, which matches Option A.