The value of ∫₀^(π/2) sin⁴ x cos² x dx is:

The value of ∫₀^(π/2) sin⁴ x cos² x dx is:

• A. π/32
• B. π/16
• C. π/64
• D. 3π/64

Answer: A) π/32

Explanation: Use Beta function or reduction. sin⁴ x cos² x. Let I = ∫₀^(π/2) sin⁴ x cos² x dx = (1/2) B(5/2, 3/2)? Actually formula: ∫₀^(π/2) sin^(2m−1) x cos^(2n−1) x dx = Γ(m)Γ(n)/(2Γ(m+n)). Here sin⁴ x = sin^(4) x, cos² x = cos^(2) x. So 2m−1=4 → m=5/2; 2n−1=2 → n=3/2. Then I = Γ(5/2)Γ(3/2) / (2 Γ(4)). Γ(5/2) = (3/2)(1/2)√π = 3√π/4. Γ(3/2) = (1/2)√π = √π/2. Γ(4) = 6. Product = (3π/8). Divide by 2×6=12 gives (3π/8)/12 = 3π/96 = π/32.