A-LevelMathematicsIntegration9709/22 · May/June 2024 · Q58 Marks

The polynomial p(x) is defined by p(x) = 9x³ +18x² +5x+4.

(a)Find the quotient when p(x) is divided by (3x+2), and show that the remainder is 6.[3]
(b)Find the value of ∫₀² p(x) / (3x+2) dx, giving your answer in the form a+lnb where a and b are integers.[5]

Mark scheme

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

[M1] Carry out division at least as far as 3x² + k₁x Or equivalent (inspection, ...).
[A1] Obtain quotient 3x² + 4x-1
[A1] Confirm remainder is 6 Answer given – necessary detail needed.
[SC B1] SC B1 for use of the factor theorem to show remainder is 6 if no other marks are awarded.
[(M1)] Synthetic division
[(A1)] Obtain quotient 3x² + 4x-1
[(A1)] Confirm remainder is 6
[B1FT] Identify integrand as 3x² + 4x-1 + 6/(3x+2) Following their quotient.
[*M1] Integrate to obtain at least x³ and k₂ ln(3x + 2) terms
[A1] Obtain x³ + 2x² − x + 2ln(3x + 2)
[DM1] Apply limits and appropriate logarithm properties
[A1] Obtain 14+ln16

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