2022HSCShort answer

Q14 (b) (iii)

2 marks

Question

Let $J_n=\int_0^1x^ne^{-x}\,dx$ , where $n$ is a non-negative integer. Show that
$J_n=nJ_{n-1}-\frac1e,\qquad n\ge1.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	2
Attempts to apply integration by parts, correctly dealing with the limits of integration, or equivalent merit	1

Sample answer:

J_n=\int_0^1x^ne^{-x}\,dx =\left[x^n(-e^{-x})\right]_0^1-\int_0^1nx^{n-1}(-e^{-x})\,dx =-\frac1e+nJ_{n-1}

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Syllabus alignment

Calculus

Further IntegrationApply substitution, partial fractions, integration by parts and recurrence relationships to theoretical and practical problems.

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