9.1 Determine the following integrals: 9.1.1 \( \int (10x + 6) \,dx \) 9.1.2 \( \int (x^2 + 2) - 2x^3 \,dx \) 9.2 The sketch below represents the shaded area bounded by the curve of the function defined by \( f(x) = \frac{3}{x} - 4; \: x > 0 \) and the x-axis between the points where \( x = 2 \) and \( x = 4 \)  Determine (showing ALL calculations) the shaded area bounded by the curve and the x-axis between the points where \( x = 2 \) and \( x = 4 \). - NSC Technical Mathematics - Question 9 - 2022 - Paper 1

To calculate ( \int (x^2 + 2 - 2x^3) ,dx ):

1. Integrate each term separately:

   • For ( x^2 ), we have ( \frac{x^3}{3} ).
   • For ( 2 ), it becomes ( 2x ).
   • For ( -2x^3 ), it becomes ( -\frac{2}{4} x^4 = -\frac{1}{2} x^4 ).

2. Combine the integrated terms: [ \frac{x^3}{3} + 2x - \frac{1}{2} x^4 + C ]

To find the shaded area bounded by the curve and the x-axis from ( x = 2 ) to ( x = 4 ), we integrate:

1. Set up the integral: [ A = \int_{2}^{4} \left( \frac{3}{x} - 4 \right) ,dx ]

2. Calculate the definite integral:

   • First, integrate: [ \int \left( \frac{3}{x} - 4 \right) ,dx = 3 \ln |x| - 4x ]
   • Evaluate from 2 to 4: [ A = \left[ 3 \ln(4) - 4(4) \right] - \left[ 3 \ln(2) - 4(2) \right] ]
   • This simplifies to: [ A = (3 \ln(4) - 16) - (3 \ln(2) - 8) ]
   • Combine and simplify further to find approximate values: [ A \approx 5.92 \text{ square units} ]