Hint
Compare the entire numerator and denominator derivative. The numerator is A times the denominator derivative.
Worked solution
(a)
Differentiate \(5 \log{\left(x^{2} + 4 x + 17 \right)}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(5 \log{\left(x^{2} + 4 x + 17 \right)} + C\)
(b)
Use upper primitive value minus lower primitive value: \(5 \log{\left(22 \right)}\) − (\(5 \log{\left(17 \right)}\)) = \(\displaystyle \log{\left(\frac{5153632}{1419857} \right)}\).
Answer: \(\displaystyle \log{\left(\frac{5153632}{1419857} \right)}\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(G(x)=5 \log{\left(x^{2} + 4 x + 17 \right)} - 5 \log{\left(17 \right)}\)
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: A constant numerator over a quadratic usually needs another pattern.
(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: Do not omit the proportionality factor.
(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: A constant numerator over a quadratic usually needs another pattern.
Total: 6 marks.