A-LevelMathematicsIntegration9709/31 · May/June 2023 · Q810 Marks

Let f(x) = (3-3x²) / ((2x + 1)(x + 2)²).

(a)Express f(x) in partial fractions.[5]
(b)Hence find the exact value of ∫₀⁴ f(x) dx, giving your answer in the form a + b ln c, where a, b and c are integers.[5]

Mark scheme

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

[B1] State or imply the form A/(2x+1) + B/(x+2) + C/(x+2)² Accept A/(2x+1) + (Dx+E)/(x+2)²
[M1] Use a correct method for finding a constant
[A1] Obtain one of A = 1, B = -2, C = 3 For alternative form: A = 1, D = -2, E = -1.
[A1] Obtain a second value
[A1] Obtain the third value
[B1 FT] Integrate and obtain one of ½ln(2x+1), – 2ln(x + 2), -3/(x+2). The follow through is on their A, B, C.
[B1 FT] Obtain a second term If the alternative form is used, then either need to use integration by parts or split the fraction further.
[B1 FT] Obtain the third term
[M1] Substitute limits correctly in an integral with at least two terms of the form ½ln(2x+1), -2ln(x + 2) and -3/(x+2) and subtract in correct order. The terms used need to have been obtained correctly. Must be exact values, not decimals.
[A1] Obtain 1 - ln 3

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