A-LevelMathematicsAlgebra9709/32 · Feb/Mar 2018 · Q810 Marks

Let f(x) = (5x² + x + 27) / ((2x + 1)(x² + 9)).

(i)Express f(x) in partial fractions.[5]
(ii)Hence find ∫₀⁴ f(x) dx, giving your answer in the form ln c, where c is an integer.[5]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2018). The mark or mark type is shown in brackets.

[B1] State or imply the form A/(2x+1) + (Bx+C)/(x²+9)
[M1] Use a correct method for finding a constant
[A1] Obtain one of A = 3, B = 1 and C = 0
[A1] Obtain a second value
[A1] Obtain the third value
[B1 FT] Integrate and obtain term ³⁄₂ln(2x+1) (FT on A value)
[M1] Integrate and obtain term of the form k ln(x² +9)
[A1 FT] Obtain term ½ln(x² +9) (FT on B value)
[M1] Substitute limits correctly in an integral of the form a ln(2x+1) + b ln(x² +9), where ab≠0
[A1] Obtain answer ln 45 after full and correct working

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Topic: Algebra, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Feb/Mar 2018 Paper 3 Variant 2, question 8.

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