Evaluate ∫₀^(π/2) x sin x dx.
Given that I_n = ∫₀^(π/2) xⁿ sin x dx, prove that, for n 1,
I_n = n(π/2)ⁿ⁻¹ – n(n - 1)I_n₋₂.
By first using the substitution x = cos⁻¹u, find the value of
∫₀¹ (cos⁻¹u)³ du,
giving your answer in an exact form.

Q: What topic does this question test?

This question tests Calculus (Integration Techniques) in A-Level Further Mathematics (syllabus code 9231). It is worth 11 marks.

Q: Which exam paper is this question from?

This question appeared in the Cambridge A-Level Further Mathematics Oct/Nov 2016 examination, Paper 1 Variant 2 (9231_w16_qp_12.pdf).

Question

Evaluate ∫₀^(π/2) x sin x dx.
Given that I_n = ∫₀^(π/2) xⁿ sin x dx, prove that, for n > 1,
I_n = n(π/2)ⁿ⁻¹ – n(n - 1)I_n₋₂.
By first using the substitution x = cos⁻¹u, find the value of
∫₀¹ (cos⁻¹u)³ du,
giving your answer in an exact form.

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Evaluate ∫₀^(π/2) x sin x dx. Given that I_n = ∫₀^(π/2) xⁿ sin x dx, prove that, for n > 1, I_n = n(π/2)ⁿ⁻¹ – n(n - 1)I_n₋₂. By first using the substitution x = cos⁻¹u, find the value of ∫₀¹ (cos⁻¹u)³ du, giving your answer in an exact form.

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