A-LevelMathematicsIntegration9709/23 · May/June 2019 · Q58 Marks

Find the quotient and remainder when 2x³ + x2 – 8x is divided by (2x + 1).

(i)Find the quotient and remainder when 2x³ + x2 – 8x is divided by (2x + 1).[3]
(ii)Hence find the exact value of ∫(from 0 to 3) (2x³ + x² – 8x) / (2x + 1) dx, giving the answer in the form ln(ke^a) where k and a are constants.[5]

Mark scheme

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

[M1] Carry out division to obtain quotient of form x² + k
[A1] Obtain quotient x² - 4
(Guidance: Allow use of an identity)
[A1] Obtain remainder 4
(Guidance: SC: If only the remainder theorem is used to obtain 4 then B1)
[*M1] Integrate to obtain at least k₁x³ and k₂ ln(2x+1) terms using the result from (i)
[A1] Obtain correct (1/3)x³ - 4x + 2ln(2x + 1)
[DM1] Apply limits and note or imply that constant k₃ can be written ln(e^k₃)
[M1] Apply appropriate logarithm properties correctly
[A1] Obtain ln(49e³)

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