Let f(x) = (5x² + x + 11) / ((4 + x²)(1 + x))

Mark scheme

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

1. [B1] State or imply the form (Ax+B)/(4+x²) + C/(1+x)
2. [M1] Use a correct method for finding a coefficient
3. [A1] Obtain one of A = 2, B = −1 and C = 3
4. [A1] Obtain a second value
5. [A1] Obtain the third value
6. [B1FT] Integrate and obtain terms B/(2) ln(4+x²) + B/2 tan⁻¹(x/2) + C ln(1+x)
7. [B1FT] Integral of Ax+B/(4+x²) is A/2 ln(4+x²) + B/2 tan⁻¹(x/2) or (A/2)ln(4 + x²) only. B0FT unless clearly from single term.
8. [M1] Substitute limits 0 and 2 correctly in an integral of the form aln(4+x²) + btan⁻¹(x/2) + cln(1+x), where abc≠0
9. [A1] AG - work to combine or simplify at least 2 of ln terms is required CWO. A0 if any non exact value(s) seen.