Hint
Do not apply the power formula at exponent −1. Increase the power by one and divide.
Worked solution
(a)
Differentiate \(x^{3} + 5 x^{2}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(x^{3} + 5 x^{2} + C\)
(b)
Use upper primitive value minus lower primitive value: \(6\) − (\(0\)) = \(6\).
Answer: \(6\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(G(x)=x^{3} + 5 x^{2}\)
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 increased exponent, not the original one.
(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: Subtract the lower primitive 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 increased exponent, not the original one.
Total: 6 marks.