Let f(x) = (-7x²+2x-6)/((1+x)(4+x²))

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2025). The mark or mark type is shown in brackets.

1. [B1] State or imply the form A/(1+x) + (Bx+C)/(4+x²)
2. [M1] Use a correct method for finding a constant Even with incorrect PF denominators A(4+x²) + (Bx + C)(1 + x) + = −7x² +2x - 6
3. [A1] Obtain one of A = −3, B = −4 and C = 6
4. [A1] Obtain a second value
5. [A1] Obtain a third value
6. [B1 FT] Obtain term -3ln(1+x) OE FT A ln (1+x)
7. [B1 FT] Obtain term -2ln(4+x²) OE FT B/2 ln (4+x²)
8. [M1] Obtain integral of the form c tan⁻¹(x/d) with d ≠ 1 following separation into two expressions d = 2 or 2 only.
9. [A1 FT] Obtain 3 tan⁻¹(x/2) FT C/2 tan⁻¹(x/2)
10. [M1] Substitute correct limits correctly in an expression (obtained correctly) of the form aln(1+x), bln(4+x²), and ctan⁻¹(x/d), where a, b, c ≠ 0. a ln(3) + b ln(8) – b ln(4) + c(¼π), where a, b, c ≠ 0. Do not allow slips, and must get to c(¼π).
11. [A1] Obtain answer ¾π - ln108 Must be in the form aπ – lnb.