Hint
Transform definite limits when changing variables. Substitute \(\displaystyle u=x^{2}+a^{2}\).
Worked solution
(a)
\(\displaystyle \frac12[\ln(x^2+49)]_0^1=0.0101013536587597\)
Answer: \(0.0101013536587597\).
Answer: \(\approx 0.0101013536587597\)
(b)
\(\displaystyle [\operatorname{arsinh}(\frac{x}{7})]_{0}^{1}= 0.142375643167804\)
Answer: \(0.142375643167804\).
Answer: \(\approx 0.142375643167804\)
(c)
\(\displaystyle [e^{2x}(\frac{x}{2}- \frac{1}{4})]_{0}^{1}= 2.09726402473266\)
Answer: \(2.09726402473266\).
Answer: \(\approx 2.09726402473266\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
- 1 mark: Substitute \(\displaystyle u=x^{2}+a^{2}\).
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not mix old limits with a new variable.
(b) · 2 marks
- 1 mark: Recognise the inverse-sinh antiderivative.
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: The subtracted integral in integration by parts has a minus sign.
(c) · 2 marks
- 1 mark: Integrate by parts with \(\displaystyle u=x\).
- 1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not mix old limits with a new variable.
Total: 6 marks.