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Integration and applications of integration — Question 15

QCAA 2025, Paper 1 · 5 marks

Q15 · 2025 · Technology-freeSimple familiar5 marks

QUESTION 15 (5 marks)

Integration by parts and area.

a)

Use integration by parts to show

\int x^2e^{-x}\,dx=-e^{-x}(x^2+2x+2)+c.

[3 marks]

b)

The area of the bounded region between the graphs of

y=x^2e^{-x}

and

y=x^2e^{-1}

over the domain

[0,1]

is given by

\int_0^1x^2\left(e^{-x}-e^{-1}\right)\,dx.

Use your result from Question 15a) to determine this area. Simplify your answer.

[2 marks]

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