The value of $\int_{0}^{\pi/2} \sin^3 x \, dx$ is:
Options:
(A) 0
(B) 1/3
(C) 2/3
(D) 1
Official Answer Key:
C
Step-by-Step Solution & Explanation
Using the standard Walker/reduction formula for ∫₀^(π/2) sinⁿ x dx with odd n=3: ∫₀^(π/2) sin³x dx = (n−1)!!/n!! = 2/3. (Alternatively, write sin³x = sinx(1−cos²x) and substitute u=cosx: ∫(u²−1)du from 1 to 0 = [u³/3−u] evaluated → gives 2/3 directly.) Hence (C) 2/3 is correct.
Section 2 — Farm Machinery
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