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Work through curated question sheets organised by exam pattern, from accessible questions to advanced challenges — so every revision session counts.
Find each integral.
Tip: write 6x² as 6x−2 first.
[Maximum mark: 4]
A curve has gradient f′(x) = 3x² − 4x + 1 and passes through (1, 5).
(a)Find an expression for f(x), including c.[2]
(b)Use the point (1, 5) to find c.[2]
Integrate term by term, then substitute x = 1, y = 5.
(a)f(x) = x³ − 2x² + x + cM1A1
M1 for raising one power, A1 all correct.
(b)5 = 1 − 2 + 1 + cM1
c = 5, so f(x) = x³ − 2x² + x + 5A1
Total [4 marks]
[Maximum mark: 8]
Let f(x) = x² − 4x + 3. The region R is enclosed by the graph of f and the x-axis.
(a)Find the x-intercepts of the graph of f.[2]
(b)Find the area of R.[3]
(c)Given a > 3, find a such that ∫3af(x) dx equals the area of R.[3]
(a)(x − 1)(x − 3) = 0M1
x = 1, x = 3A1
(b)∫13(x² − 4x + 3) dxM1
[x³3 − 2x² + 3x] = −43A1
area = 43A1
(c)a³3 − 2a² + 3a = 43M1
(a − 1)²(a − 4) = 0A1
a = 4A1
Total [8 marks]
WorksheetsStep by step, up to exam level.One topic climbs from warm-up integrals to a full exam-style question — with a marked answer key at every step.
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