A-LevelMathematicsIntegration9709/22 · Oct/Nov 2023 · Q58 Marks

Find the quotient when 6x³ – 5x² – 24x – 4 is divided by (2x + 1), and show that the remainder is 6.

(a)Find the quotient when 6x³ – 5x² – 24x – 4 is divided by (2x + 1), and show that the remainder is 6.[3]
(b)Hence find ∫[from 2 to 7] (6x³ - 5x² – 24x – 4) / (2x + 1) dx, giving your answer in the form a + ln b, where a and b are integers.[5]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2023). The mark or mark type is shown in brackets.

[M1] (a) Carry out division at least as far as 3x² + k₁x OE (e.g. by inspection).
[A1] (a) Obtain quotient 3x² -4x-10
[A1] (a) Confirm given result of remainder is 6 with sufficient detail AG SC If remainder = 6 shown using remainder theorem allow B1.
[*M1] (b) Integrate to obtain at least x³ and term of form k₂ ln(2x+1) In term must be added.
[A1] (b) Obtain x³ - 2x² -10x+3ln(2x+1)
[DM1] (b) Apply limits correctly to expression with four terms
[DM1] (b) Apply appropriate logarithm properties correctly to obtain the form k₃ ln a
[A1] (b) Obtain 195+ln 27

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