AS Level · Mathematics 9709 · Pure Mathematics 2

Integration: AS Level Mathematics questions with answers

Free Integration questions for AS Level Mathematics (9709), part of Pure Mathematics 2. Each question has the answer and the marking points an examiner would look for.

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Integration practice questions

Question 1Structured · 5 marksA* challenge
(a)

Use the trapezium rule with 3 intervals to estimate ∫01 √(1 + x3) dx. Give your answer correct to 3 decimal places.

[3]
(b)

Explain, with reference to the shape of the curve, whether your answer to (a) is an over-estimate or an under-estimate.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1.121
h = 0.333 B1
correct trapezium formula with y-values M1
1.121 A1
3
(b)the curve is convex (concave upwards) on this interval, so the chords lie above the curve
curve concave up / convex B1
trapezia lie above the curve B1
2
Question 2Structured · 3 marks

Find ∫ 4x + 1 dx and hence find the exact value of ∫01 4x + 1 dx.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
4 ln(x + 1) + c; 4 ln 2
ln form M1
4 ln(x + 1) A1
4 ln 2 A1
3
Question 3Structured · 3 marks

Find the exact value of ∫ from 0 to 1 of e3x dx.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
13(e3 − 1)
13e3x M1
Correct limits M1
13(e3 − 1) A1
3
Question 4Structured · 5 marksA* challenge
(a)

Use the trapezium rule with 2 intervals to estimate ∫01 √(1 + x3) dx. Give your answer correct to 3 decimal places.

[3]
(b)

Explain, with reference to the shape of the curve, whether your answer to (a) is an over-estimate or an under-estimate.

[2]
Show answers and mark scheme
PartAnswer and marking pointsMarks
(a)1.134
h = 0.5 B1
correct trapezium formula with y-values M1
1.134 A1
3
(b)the curve is convex (concave upwards) on this interval, so the chords lie above the curve
curve concave up / convex B1
trapezia lie above the curve B1
2
Question 5Structured · 3 marks

Find ∫ 33x + 1 dx and hence find the exact value of ∫01 33x + 1 dx.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
1 ln(3x + 1) + c; 1 ln 4
ln form M1
1 ln(3x + 1) A1
1 ln 4 A1
3
Question 6Structured · 3 marks

Find the exact value of ∫ from 0 to 2 of e3x dx.

[3]
Show answers and mark scheme
PartAnswer and marking pointsMarks
13(e6 − 1)
13e3x M1
Correct limits M1
13(e6 − 1) A1
3

More Pure Mathematics 2 questions

AlgebraLogarithmic & exponential functionsTrigonometryDifferentiationNumerical solution of equations

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Original Cambridge-style practice questions written for mentorriyaz.in. These are not Cambridge past paper questions, and this site is not affiliated with or endorsed by Cambridge University Press & Assessment. Updated 2026-11-04. Paper generator · mentorriyaz.in