A-LevelMathematicsIntegration9709/32 · Feb/Mar 2023 · Q89 Marks
The diagram shows the curve y = x³ ln x, for x > 0, and its minimum point M. [Figure 8.1]
- (a)Find the exact coordinates of M.[4]
- (b)Find the exact area of the shaded region bounded by the curve, the x-axis and the line x = ½.[5]
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Mark scheme
Marking points from the official mark scheme (Feb/Mar 2023). The mark or mark type is shown in brackets.
- [*M1] Use the product rule correctly
- [A1] Obtain the correct derivative in any form e.g. x³/x + 3x² ln x .
- [DM1] Equate derivative to zero and solve exactly for x
- [A1] Obtain answer (e⁻¹ᐟ³, -1/(3e)) or exact equivalent ISW
- [*M1] Integrate by parts and reach ax⁴ lnx + b∫(x⁴ / x)dx
- [A1] Obtain x⁴/4 lnx - ∫(x³/4)dx OE
- [A1] Complete integration and obtain x⁴/4 lnx - x⁴/16 OE
- [DM1] Use limits of x = 1/2 and x = 1 in the correct order, having integrated twice
- [A1] Obtain answer 15/256 - 1/64 ln 2 or exact equivalent final answer
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