Hint
Complete any square before choosing a standard form. Scale x by b in the inverse tangent derivative.
Worked solution
(a)
Differentiate \(\displaystyle \frac{2 \operatorname{atan}{\left(\frac{x}{3} \right)}}{3}\) to recover the integrand. Include an arbitrary constant on each interval of the domain.
Answer: \(\displaystyle \frac{2 \operatorname{atan}{\left(\frac{x}{3} \right)}}{3} + C\)
(b)
Use upper primitive value minus lower primitive value: \(\displaystyle \frac{\pi}{6}\) − (\(0\)) = \(\displaystyle \frac{\pi}{6}\).
Answer: \(\displaystyle \frac{\pi}{6}\)
(c)
Subtract the primitive value at the initial abscissa, so the stated initial condition is satisfied.
Answer: \(\displaystyle G(x)=\frac{2 \operatorname{atan}{\left(\frac{x}{3} \right)}}{3}\)
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 over \(\displaystyle x^{2}+b^{2}\) is not simply a logarithmic derivative.
(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 factor \(\displaystyle \frac{1}{b}\) remains outside arctan.
(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 over \(\displaystyle x^{2}+b^{2}\) is not simply a logarithmic derivative.
Total: 6 marks.