A-LevelMathematicsIntegration9709/32 · Feb/Mar 2024 · Q1010 Marks

Let `f(x) = 36a² / ((2a+x)(2a-x)(5a-2x))`, where a is a positive constant.

(a)Express f(x) in partial fractions.[5]
(b)Hence find the exact value of `∫(from -a to a) f(x)dx`, giving your answer in the form `plnq + rlns` where p and r are integers and q and s are prime numbers.[5]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2024). The mark or mark type is shown in brackets.

[B1] State or imply the form A/(2a+x) + B/(2a-x) + C/(5a-2x)
[M1] Use a correct method for finding a coefficient
[A1] Obtain one of A = 1, B = 9, C = -16
[A1] Obtain a second value
[A1] Obtain the third value
[B1 FT] Integrate and obtain one of the terms ln|2a+x|, -9ln|2a-x|, 8ln|5a-2x|
[B1 FT] Obtain a second correct term
[B1 FT] Obtain the third correct term
[M1] Substitute limits correctly in an integral of the form pln|2a+x|+qln|2a-x|++rln|5a−2x| and remove all a's
[A1] Obtain 18ln3-8ln7 from correct working

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Feb/Mar 2024 Paper 3 Variant 2, question 10.

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