The diagram shows the graph of y = e^(cosx) sin³x for 0 ≤ x ≤ π, and its maximum point M. The shaded region R is bounded by the curve and the x-axis.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [M1] Use product rule and chain rule at least once
2. [A1] Obtain correct derivative in any form
3. [M1] Equate derivative to zero, use Pythagoras and obtain an equation in cos x
4. [A1] Obtain cos²x + 3cosx−1=0, or 3-term equivalent
5. [A1] Obtain answer x = 1.26
6. [M1] Using du = ± sin x dx express integrand in terms of u and du
7. [A1] Obtain integrand eᵘ (u²-1)
8. [*M1] Commence integration by parts and reach aeᵘ (u²-1)+b∫ueᵘ du
9. [A1] Obtain eᵘ (u²-1)-2∫ueᵘ du
10. [A1] Complete integration, obtaining eᵘ (u² – 2u +1)
11. [DM1] Substitute limits u = 1 and u = − 1 (or x = 0 and x = π), having integrated completely
12. [A1] Obtain answer 4/e, or exact equivalent