The diagram shows the curve y = (1 + x²)e⁻ˣ/² for x ≥ 0. The shaded region R is enclosed by the curve, the x-axis and the lines x = 0 and x = 2. [Figure 9.1]

Mark scheme

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

1. [M1] (i) Use correct product or quotient rule
2. [A1] (i) Obtain correct derivative in any form
3. [M1] (i) Equate derivative to zero and obtain a 3 term quadratic equation in x
4. [A1] (i) Obtain answers x = 2 ± √3
5. [*M1] (ii) Integrate by parts and reach k(1+x²)e⁻ˣ/² + ∫ xe⁻ˣ/² dx
6. [A1] (ii) Obtain -2(1+x²)e⁻ˣ/² + 4∫ xe⁻ˣ/² dx, or equivalent
7. [A1] (ii) Complete the integration and obtain (-18 - 8x - 2x²)e⁻ˣ/², or equivalent
8. [DM1] (ii) Use limits x = 0 and x = 2 correctly, having fully integrated twice by parts
9. [A1] (ii) Obtain the given answer