IB · Mathematics: Analysis & Approaches SL & HL · Calculus

Integration: IB Mathematics: Analysis & Approaches questions with answers

Free Integration questions for IB Mathematics: Analysis & Approaches (SL & HL), part of Calculus. 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

Given that ∫1k(4x − 4)dx = 32, where k > 1, find the value of k.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
k = 5
[2x2 − 4x]1k A1
2k2 − 4k − (−2) = 32 M1
2k2 − 4k − 30 = 0 A1
attempt to solve (M1)
k = 5 (reject −3 since k > 1) A1
5
Question 2Structured · 5 marks

The gradient of a curve is given by dydx = 12x(x2 + 4)2. The curve passes through the point (1, 1).

Find the equation of the curve.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
y = 2(x2 + 4)3 − 249
attempt to integrate using substitution or reverse chain rule M1
2(x2 + 4)3 (+ C) A2
substituting the point to find C M1
C = −249 A1
5
Question 3Structured · 4 marks

Find the exact value of ∫03xx2 + 1dx.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
12ln 10
recognising ∫f′(x)f(x)dx form (M1)
12ln(x2 + 1) A1
substituting limits (M1)
12ln 10 A1
4
Question 4Structured · 5 marksA* challenge

Given that ∫0k(2x + 1)dx = 30, where k > 0, find the value of k.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
k = 5
[x2 + x]0k A1
k2 + k − (0) = 30 M1
k2 + k − 30 = 0 A1
attempt to solve (M1)
k = 5 (reject −6 since k > 0) A1
5
Question 5Structured · 5 marks

The gradient of a curve is given by dydx = 18x(x2 − 1)2. The curve passes through the point (0, 5).

Find the equation of the curve.

[5]
Show answers and mark scheme
PartAnswer and marking pointsMarks
y = 3(x2 − 1)3 + 8
attempt to integrate using substitution or reverse chain rule M1
3(x2 − 1)3 (+ C) A2
substituting the point to find C M1
C = 8 A1
5
Question 6Structured · 4 marks

Find the exact value of ∫01xx2 + 4dx.

[4]
Show answers and mark scheme
PartAnswer and marking pointsMarks
12ln 54
recognising ∫f′(x)f(x)dx form (M1)
12ln(x2 + 4) A1
substituting limits (M1)
12ln 54 A1
4

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Original IB-style practice questions written for mentorriyaz.in. These are not IB past paper questions, and this site is not affiliated with or endorsed by the International Baccalaureate Organization. Updated 2026-11-04. Paper generator · mentorriyaz.in