Hint
Double the angle inside the cosine. Combine the constant halves in the double-angle identities.
Worked solution
(a)
Differentiate \(\displaystyle 7 x + \frac{\sin{\left(4 x \right)}}{2}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(\displaystyle 7 x + \frac{\sin{\left(4 x \right)}}{2} + C\)
(b)
Use upper primitive value minus lower primitive value: \(\displaystyle \frac{7 \pi}{4}\) − (\(0\)) = \(\displaystyle \frac{7 \pi}{4}\).
Answer: \(\displaystyle \frac{7 \pi}{4}\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(\displaystyle G(x)=7 x + \frac{\sin{\left(4 x \right)}}{2}\)
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 minus sign belongs to the sine-squared identity.
(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 apply the ordinary power rule to \(\displaystyle \sin ^{2}x\).
(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 minus sign belongs to the sine-squared identity.
Total: 6 marks.