Hint
Transform definite limits when changing variables. Substitute u=x²+a².
Worked solution
(a)
\(\frac12[\ln(x^2+49)]_0^1=0.0101013536587597\)
Answer: \(0.0101013536587597\).
Answer: \(\approx 0.0101013536587597\)
(b)
\([\operatorname{arsinh}(x/7)]_0^1=0.142375643167804\)
Answer: \(0.142375643167804\).
Answer: \(\approx 0.142375643167804\)
(c)
\([e^{2x}(x/2-1/4)]_0^1=2.09726402473266\)
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 u=x²+a².
- 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 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.