2022HSCShort answer

Q14 (b) (i)

1 mark

Question

Let
$J_n=\int_0^1x^ne^{-x}\,dx,$
where $n$ is a non-negative integer. Show that
$J_0=1-\frac1e.$

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	1

Sample answer:
$J_n=\int_0^1x^ne^{-x}\,dx,\quad n\ge0$

n=0:\quad J_0=\int_0^1e^{-x}\,dx=-[e^{-x}]_0^1=-\left(\frac1e-1\right)=1-\frac1e

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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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