A-LevelMathematicsIntegration9709/31 · May/June 2024 · Q87 Marks

Use the substitution u = 1-sinx to find the exact value of integral from π/6 to (3/2)π of (sin(2x) / sqrt(1-sinx)) dx. Give your answer in the form a+b√2 where a and b are rational numbers to be determined.

Use the substitution u = 1-sinx to find the exact value of integral from π/6 to (3/2)π of (sin(2x) / sqrt(1-sinx)) dx. Give your answer in the form a+b√2 where a and b are rational numbers to be determined.[7]

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

[B1] State or imply du = – cosx dx
[M1] Use sin 2x = 2sinxcosx and write the integral in terms of u
[A1] Obtain ±2 ∫ (1-u)/√u du or equivalent
[DM1] Integrate correctly to obtain au^(1/2) + bu^(3/2)
[A1] Obtain correct -4u^(1/2) + (4/3)u^(3/2)
[DM1] Correctly use limits u = 2 and 0 in an expression of the form au^(1/2) + bu^(3/2) OR limits x = π/6 and π/2 in an expression of the form a(1-sinx)^(1/2)+b(1-sinx)^(3/2)
[A1] Obtain answer -5/3 √2

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Topic: Integration, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source: A-Level Mathematics, May/June 2024 Paper 3 Variant 1, question 8.

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