A-LevelMathematicsIntegration9709/32 · Oct/Nov 2022 · Q1010 Marks

Let f(x) = (4 – x + x²) / ((1+x)(2 + x²))

(a)Express f(x) in partial fractions.[5]
(b)Find the exact value of ∫(from 0 to 4) f(x) dx. Give your answer as a single logarithm.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2022). The mark or mark type is shown in brackets.

[B1] (a) State or imply the form A/(1+x) + (Bx+C)/(2+x²)
[M1] (a) Use a correct method for finding a constant
[A1] (a) Obtain one of A = 2, B = -1 and C = 0
[A1] (a) Obtain a second value
[A1] (a) Obtain the third value
[B1 FT] (b) Integrate and obtain term 2ln(1+x)
[*M1] (b) Integrate and obtain term of the form kln(2+x²) from an integral of the correct form
[A1 FT] (b) Obtain term -½ln(2+x²)
[DM1] (b) Substitute limits in an integral containing terms of the form aln(1+x)+bln(2+x²), where ab ≠ 0
[A1] (b) Obtain answer ln(25/3)

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Oct/Nov 2022 Paper 3 Variant 2, question 10.

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