Bepaal die volgende: 9.1.1 \[ \int 2x \; dx \] 9.1.2 \[ \int \left( \sqrt{x} + \frac{7}{x} + 4x^{-2} \right) dx \] 9.2 Die skets hieronder verteenwoordig funksie \( f \) wat deur \( f(x) = -x^2 - x + 6 \) gedefinieer word, met \( x \)-asfinte by \( (-3; 0) \) en \( (2; 0) \) - NSC Technical Mathematics - Question 9 - 2020 - Paper 1

This integral can be split up and handled term by term:

[ \int \left( \sqrt{x} + \frac{7}{x} + 4x^{-2} \right) dx = \int x^{1/2} , dx + \int 7x^{-1} , dx + \int 4x^{-2} , dx ]

Calculating each term:

1. ( \int x^{1/2} , dx = \frac{x^{3/2}}{3/2} = \frac{2}{3}x^{3/2} )
2. ( \int 7x^{-1} , dx = 7 \ln|x| )
3. ( \int 4x^{-2} , dx = 4 \cdot \left( -\frac{1}{x} \right) = -\frac{4}{x} )

Combining these, we have:

[ \int \left( \sqrt{x} + \frac{7}{x} + 4x^{-2} \right) dx = \frac{2}{3}x^{3/2} + 7 \ln|x| - \frac{4}{x} + C ]

To find the unshaded area bounded by the curve and the x-axis, we first compute the definite integral of the function from ( x = -3 ) to ( x = 2 ):

[ \int_{-3}^{2} (-x^2 - x + 6) , dx ]

Calculating this integral:

1. Find the antiderivative: [ \int (-x^2 - x + 6) , dx = -\frac{x^3}{3} - \frac{x^2}{2} + 6x + C ]

2. Evaluate from -3 to 2: [ \left[ -\frac{(2)^3}{3} - \frac{(2)^2}{2} + 6(2) \right] - \left[ -\frac{(-3)^3}{3} - \frac{(-3)^2}{2} + 6(-3) \right] ] [ = \left[ -\frac{8}{3} - 2 + 12 \right] - \left[ 9 - \frac{9}{2} - 18 \right] ] Simplifying this gives: [ = \left[ \frac{28}{3} \right] - \left[ -\frac{27}{2} \right] ] The total area is to be computed considering the unshaded area is less than the shaded area.