Find the exact value of ∫[from 0 to π/2] x² cos(x/3)dx.

Mark scheme

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

1. [*M1] Commence integration by parts and reach Ax² sin(x/3) + ∫Bx sin(x/3)dx OE BOD on ± otherwise scores 0/6.
2. [A1 OE] Obtain 3x² sin(x/3) - ∫6x sin(x/3)dx Allow 3×2 for 6.
3. [*DM1] Complete integration by parts and reach Ax² sin(x/3) + Bx cos(x/3) + C sin(x/3) OE
4. [A1 OE] Obtain 3x² sin(x/3) + 18x cos(x/3) - 54 sin(x/3) oe Allow 6 × 3 for 18, 9 × 6 for 54 OE.
5. [DM1] Substitute limits correctly in an expression of the form Ax² sin(x/3) + Bx cos(x/3) + C sin(x/3), where ABC ≠ 0 Need to use sin(π/6) = ½ and cos(π/6) = √3/2 to obtain Aπ²√3/2 + Bπ√3/2 + C√3/2
6. [A1] Obtain answer (3√3π²/2) + 9π - 27√3 or exact equivalent ISW Allow √27 for 3√3, 18 for 9, 54 for 27, √2187 for 27√3 etc.