It is given that ∫(1 to a) x² ln x dx = 2, where a > 1.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

1. [1] Integrate by parts and reach ax³ ln x + b∫ x² (1/x) dx
2. [1] Obtain (1/3)x³ ln x - ∫ (1/3)x² dx
3. [1] Obtain integral (1/3)x³ ln x - (1/9)x³, or equivalent
4. [1] Substitute limits correctly and equate to 2
5. [1] Obtain the given answer correctly
6. [1] Evaluate a relevant expression or pair of expressions at x = 2 and x = 4
7. [1] Complete the argument correctly with correct calculated values
8. [1] Use the iterative formula correctly at least once
9. [1] Obtain final answer 3.031
10. [1] Show sufficient iterations to 5 d.p. to justify 3.031 to 3 d.p., or show there is a sign change in the interval (3.0305, 3.0315)