Let In = ∫ (from 0 to (π/2)) cosⁿ x sin²x dx, for n ≥ 0. By differentiating cosⁿ⁻¹ x sin³ x with respect to x, prove that (n + 2)In = (n − 1)In−2 for n ≥ 2.
Hence find the exact value of I₄.

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Let In = ∫ (from 0 to (π/2)) cosⁿ x sin²x dx, for n ≥ 0. By differentiating cosⁿ⁻¹ x sin³ x with respect to x, prove that (n + 2)In = (n − 1)In−2 for n ≥ 2. Hence find the exact value of I₄.

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