Let f(x) = (6x² + 8x + 9) / ((2 − x)(3 + 2x)²).

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.

1. [B1] (i) State or imply the form A/(2-x) + B/(3+2x) + C/(3+2x)²
2. [M1] (i) Use a correct method to find a constant
3. [A1] (i) Obtain one of A = 1, B = -1, C = 3
4. [A1] (i) Obtain a second value
5. [A1] (i) Obtain the third value
6. [Mark the form A/(2-x) + (Dx + E)/(3+2x)² where A = 1, D = -2 and E = 0, B1M1A1A1A1 as above.]
7. [B3ft] (ii) Integrate and obtain terms -ln(2-x) - 1/2 ln(3+2x) - 3/(2(3+2x))
8. [M1] (ii) Substitute correctly in an integral with terms a ln (2 – x), b ln (3 + 2x) and c / (3 + 2x) where abc ≠ 0
9. [A1] (ii) Obtain the given answer after full and correct working
10. [Correct integration of the A, D, E form gives an extra constant term if integration by parts is used for the second partial fraction.]