<p>Evaluate the following: <br> 17. \(\int_0^{\pi/5} \cos^2\frac{x}{2} \cdot \sin x \, dx\)</p>
Step-by-Step Solution
Key Concept: Substitute u = cos(x/2) to convert the integrand into a polynomial form, reducing the complexity from trigonometric to algebraic integration.
<p><strong>Step 1:</strong> Rewrite sin(x) using double angle formula.</p><p>sin(x) = 2sin(x/2)cos(x/2)</p><p>So the integral becomes: ∫₀^(π/5) cos²(x/2) · 2sin(x/2)cos(x/2) dx = ∫₀^(π/5) 2cos³(x/2)sin(x/2) dx</p><p><strong>Step 2:</strong> Apply substitution u = cos(x/2).</p><p>Then du = -½sin(x/2) dx, so sin(x/2) dx = -2du</p><p>When x = 0: u = cos(0) = 1</p><p>When x = π/5: u = cos(π/10)</p><p><strong>Step 3:</strong> Transform the integral.</p><p>∫₀^(π/5) 2cos³(x/2)sin(x/2) dx = ∫₁^(cos π/10) 2u³ · (-2du) = -4∫₁^(cos π/10) u³ du</p><p><strong>Step 4:</strong> Integrate.</p><p>= -4[u⁴/4]₁^(cos π/10) = -[u⁴]₁^(cos π/10) = -[cos⁴(π/10) - 1] = 1 - cos⁴(π/10)</p><p>∴ <strong>Answer: 1 - cos⁴(π/10)</strong></p><p>Or equivalently: <strong>sin²(π/10)(2 - sin²(π/10))</strong> or <strong>4/5</strong> if cos(π/10) = (1+√5)/4</p>
Correct Answer: 1