5 Find the exact value of ∫(from 0 to 6) [x(x + 1) / (x² + 4)] dx.

Mark scheme

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

1. B1 Split fraction to obtain 1 + (x-4)/(x²+4)
2. M1 Attempt integration and obtain p ln(x²+4) or q tan⁻¹(x/2) from correct working
3. A1 FT Obtain ½ ln(x²+4) Follow through is on their coefficients in the partial fraction. Allow from x/(x²+4) - x/(x²+4) even if the split of the fraction is not complete. If 1 - x/(x²+4) + 4/(x²+4) later seen or implied, award the B1. Only available from a correct split, not from an approach using parts that is incomplete.
4. A1 FT Obtain -2 tan⁻¹(x/2) Only available from a correct split, not from an approach using parts that is incomplete.
5. M1 Correct use of correct limits 0 and 6 in an expression involving p ln(x²+4), q tan⁻¹(x/2) and no incorrect terms.
6. A1 Obtain 6 + ½ ln 10 – 2 tan⁻¹ 3 ISW Or three term equivalent. (Must combine the ln terms.) Accept with ln|10|.