A-LevelMathematicsIntegration9709/32 · May/June 2020 · Q69 Marks

The diagram shows the curve y = x / (1 + 3x⁴) for x ≥ 0, and its maximum point M. [Figure 6.1]

(a)Find the x-coordinate of M, giving your answer correct to 3 decimal places.[4]
(b)Using the substitution u = √3x², find by integration the exact area of the shaded region bounded by the curve, the x-axis and the line x = 1.[5]

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Mark scheme

Marking points from the official mark scheme (May/June 2020). The mark or mark type is shown in brackets.

[M1] Use quotient or product rule
[A1] Obtain correct derivative in any form e.g. ( (1+3x⁴) - x(12x³) ) / (1+3x⁴)²
[M1] Equate derivative to zero and solve for x
[A1] Obtain answer 0.577
[B1] State or imply du = 2√3x dx, or equivalent
[M1] Substitute for x and dx
[A1] Obtain integrand 1 / (2√3(1+u²)), or equivalent
[M1] State integral of the form a tan⁻¹ u and use limits u = 0 and u = √3 (or x = 0 and x = 1) correctly
[A1] Obtain answer √3π/18, or exact equivalent

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