Let f(x) = (10x + 9) / ((2x + 1)(2x + 3)²).

Mark scheme

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

1. [B1] State or imply the form A/(2x+1) + B/(2x+3) + C/(2x+3)²
2. [M1] Use a correct method to find a constant
3. [A1 A1 A1] Obtain the values A = 1, B = − 1, C = 3 (Full marks for the three correct constants – do not actually need to see the partial fractions. [Mark the form A/(2x+1) + (Dx+E)/(2x+3)² where A=1, D=-2 and E=0, B1M1A1A1A1 as above.])
4. [B1 B1 B1] Integrate and obtain terms ½ln(2x+1) - ½ln(2x+3) - 3/(2(2x+3)) (FT on A, B and C. [Correct integration of the A, D, E form of fractions gives ½ln(2x+1) + (-2)/(2x+3) - ½ln(2x+3) if integration by parts is used for the second partial fraction.])
5. [M1] Substitute limits correctly in an integral with terms a ln (2x+1), b ln (2x+3) and c/(2x+3), where abc ≠ 0 (If using alternative form: cx / (2x + 3)) (value for upper limit – value for lower limit. 1 slip in substituting can still score M1. Condone omission of ln(1))
6. [A1] Obtain the given answer following full and correct working (AG)