9.1 Bepaal die volgende integrale: 9.1.1 $\, \int (10x + 6) \, dx$ 9.1.2 $\, \int (x^2 + 2) - 2x^3 \, dx$ 9.2 Die skets hieronder verteenwoordig die geareaerde oppervlakte begrens deur die kurwe van die funksie gedefinieer deur $f(x) = \frac{3}{x} - 4$; $x > 0$ en die x-as tussen die punte waar $x = 2$ en $x = 4$ - NSC Technical Mathematics - Question 9 - 2022 - Paper 1

For this integral, we can also apply the power rule:

$\int (x^2 + 2 - 2x^3) \, dx = \frac{x^3}{3} + 2x - \frac{2x^4}{4} + C = \frac{x^3}{3} + 2x - \frac{x^4}{2} + C$

To find the shaded area between the curve and the x-axis from $x = 2$ to $x = 4$, we compute:

$A = \int_{2}^{4} \left(\frac{3}{x} - 4\right) \, dx$

Evaluating the integral gives:

$A = [3 \ln x - 4x]_{2}^{4} = [3 \ln(4) - 4(4)] - [3 \ln(2) - 4(2)]$

Computing this results in:

$A \approx 5.92 \text{ square units}$