Hint
Transform the differential and every factor. Use du=2x dx and integrate \(\displaystyle u^{n}\).
Worked solution
(a)
Differentiate \(\displaystyle \frac{\left(x^{2} + 1\right)^{4}}{4}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(\displaystyle \frac{\left(x^{2} + 1\right)^{4}}{4} + C\)
(b)
Use upper primitive value minus lower primitive value: \(4\) − (\(\displaystyle \frac{1}{4}\)) = \(\displaystyle \frac{15}{4}\).
Answer: \(\displaystyle \frac{15}{4}\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(\displaystyle G(x)=\frac{\left(x^{2} + 1\right)^{4}}{4} - \frac{1}{4}\)
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 missing factor in du changes the whole integral.
(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 reuse x-limits on a u-expression.
(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 missing factor in du changes the whole integral.
Total: 6 marks.