Hint
The permitted index range is stated. Integrate the exponential directly.
Worked solution
(a)
\(\displaystyle I_{0}= \frac{e^{2}- 1}{2}= 3.19452804946533\)
Answer: \(3.19452804946533\).
Answer: \(\approx 3.19452804946533\)
(b)
Take \(u=x^n,\ dv=e^{2x}dx\). Integration by parts gives \(\displaystyle I_{n}= [\frac{x^{n}e^{2x}}{2}]_{0}^{1}- (\frac{n}{2})I_{n- 1}= \frac{e^{2}}{2}- (\frac{n}{2})I_{n- 1}\) for \(\displaystyle n\ge 1\). Start with \(\displaystyle I_{0}= \frac{e^{2}- 1}{2}\) and repeat to obtain \(1.29863201236633\).
Answer: \(\approx 1.29863201236633\)
(c)
\(\displaystyle I_{4}= \frac{e^{2}}{2}- (\frac{4}{2})I_{3}= 1.09726402473266\)
Answer: \(1.09726402473266\).
Answer: \(\approx 1.09726402473266\)
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: Integrate the exponential directly.
- 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 apply an \(\displaystyle n\ge 1\) formula to \(\displaystyle n=0\) without checking it.
(b) · 4 marks
- 1 mark: Choose integration by parts with the power as u and exponential as dv.
- 1 mark: Evaluate both boundary terms on the stated index range.
- 1 mark: Derive the valid recurrence and use its base integral.
- 1 mark: Evaluate the requested indexed integral correctly.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: A recurrence needs a base integral.
(c) · 2 marks
- 1 mark: Apply the recurrence once more.
- 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 apply an \(\displaystyle n\ge 1\) formula to \(\displaystyle n=0\) without checking it.
Total: 8 marks.