The value of ∫₀^π x sin³ x dx is:
The value of ∫₀^π x sin³ x dx is:
- A. 0
- B. π/2
- C. 2π/3
- D. 4π/3
Answer: C) 2π/3
Explanation: Use property: ∫₀^π x f(sin x) dx = (π/2) ∫₀^π f(sin x) dx. Here f(sin x) = sin³ x. So I = (π/2) ∫₀^π sin³ x dx. Since sin³ x is odd about π/2? Actually ∫₀^π sin³ x dx = 2 ∫₀^(π/2) sin³ x dx = 2 × (2/3) = 4/3. So I = (π/2) × (4/3) = 2π/3.
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