A-LevelMathematicsIntegration9709/31 · May/June 2017 · Q1012 Marks
The diagram shows the curve y = sin x cos² 2x for 0 ≤ x ≤ ¼π and its maximum point M. [Figure X.X]
- (i)Using the substitution u = cos x, find by integration the exact area of the shaded region bounded by the curve and the x-axis.[6]
- (ii)Find the x-coordinate of M. Give your answer correct to 2 decimal places.[6]
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Mark scheme
Marking points from the official mark scheme (May/June 2017). The mark or mark type is shown in brackets.
- State or imply du = -sin x dx B1
- Using correct double angle formula, express the integral in terms of u and du M1
- Obtain integrand ±(2u²-1)² A1
- Change limits and obtain correct integral ∫(2u² – 1)² du with no errors seen A1
- Substitute limits in an integral of the form au⁵ + bu³ + cu M1
- Obtain answer ⅓(7-4√2), or exact simplified equivalent A1
- Use product rule and chain rule at least once M1
- Obtain correct derivative in any form A1
- Equate derivative to zero and use trig formulae to obtain an equation in cos x and sin x M1
- Use correct methods to obtain an equation in cos x or sin x only M1
- Obtain 10cos²x = 9 or 10sin²x=1, or equivalent A1
- Obtain answer 0.32 A1
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