Hint
Divide by the inner derivative and the increased power. Divide by a(n+1).
Worked solution
(a)
Differentiate \(\displaystyle \frac{\left(2 x + 3\right)^{4}}{2}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(\displaystyle \frac{\left(2 x + 3\right)^{4}}{2} + C\)
(b)
Use upper primitive value minus lower primitive value: \(\displaystyle \frac{2401}{2}\) − (\(\displaystyle \frac{81}{2}\)) = \(1160\).
Answer: \(1160\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(\displaystyle G(x)=\frac{\left(2 x + 3\right)^{4}}{2} - \frac{81}{2}\)
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: The integration constant need not be zero.
(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: The exponent −1 uses a logarithm instead.
(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: The integration constant need not be zero.
Total: 6 marks.