A-LevelMathematicsIntegration9709/32 · Oct/Nov 2019 · Q810 Marks

Let f(x) = (2x² + x + 8) / ((2x−1)(x² + 2)).

(i)Express f(x) in partial fractions.[5]
(ii)Hence, showing full working, find ∫(from 1 to 5) f(x) dx, giving the answer in the form ln c, where c is an integer.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[B1] State or imply the form A/(2x-1) + (Bx+C)/(x²+2)
[M1] Use a correct method for finding a constant
[A1] Obtain one of A = 4, B = −1, C = 0
[A1] Obtain a second value
[A1] Obtain the third value
[B1FT] Integrate and obtain term 2ln(2x−1)
[*M1] Integrate and obtain term of the form kln(x²+2)
[A1FT] Obtain term -(1/2)ln(x²+2)
[DM1] Substitute limits correctly in an integral of the form aln(2x-1)+bln(x²+2), where ab ≠ 0
[A1] Obtain answer ln 27 after full and correct exact working

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Oct/Nov 2019 Paper 3 Variant 2, question 8.

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