Hint
Exclude the pole from the interval. Divide by b.
Worked solution
(a)
Differentiate \(3 \log{\left(x + 2 \right)}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(3 \log{\left(x + 2 \right)} + C\)
(b)
Use upper primitive value minus lower primitive value: \(3 \log{\left(3 \right)}\) − (\(3 \log{\left(2 \right)}\)) = \(\displaystyle \log{\left(\frac{27}{8} \right)}\).
Answer: \(\displaystyle \log{\left(\frac{27}{8} \right)}\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(G(x)=3 \log{\left(x + 2 \right)} - 3 \log{\left(2 \right)}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Choose and correctly apply a valid integration method, including inner scaling.
- 1 mark: Give a complete equivalent primitive with an arbitrary constant.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Divide by the inner coefficient b.
(b) · 2 marks
- 1 mark: Apply the limits to a valid primitive in the correct order.
- 1 mark: Give the complete exact definite-integral value.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A real logarithm requires a positive argument or an absolute value.
(c) · 2 marks
- 1 mark: Find a primitive and determine its constant from the given initial value.
- 1 mark: Give the complete function G satisfying both its derivative and initial condition.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Divide by the inner coefficient b.
Total: 6 marks.