Definite Integration
Evaluation of definite integrals
Grade 12
Question:
<p>The value of \(\int_0^{\pi} \sqrt{1 + 4\sin^2\dfrac{x}{2} - 4\sin\dfrac{x}{2}}\,dx\) is:</p>
<p>\(4\sqrt{3} - \dfrac{\pi}{3} - 4\)</p>
<p>\(4\sqrt{3} - \dfrac{2\pi}{3} - 4\)</p>
<p>\(4\sqrt{3} + \dfrac{\pi}{3} - 4\)</p>
<p>\(\pi - 4\)</p>
Step-by-Step Solution
Key Concept: Recognize that the expression under the square root is a perfect square: 1 + 4sin²(x/2) - 4sin(x/2) = (1 - 2sin(x/2))². This simplifies the integral dramatically by eliminating the square root.
<p><strong>Step 1:</strong> Recognize the perfect square under the radical.</p><p>Let u = 1 - 2sin(x/2). Then:</p><p>u² = 1 - 4sin(x/2) + 4sin²(x/2) = 1 + 4sin²(x/2) - 4sin(x/2) ✓</p><p><strong>Step 2:</strong> Simplify the integral.</p><p>∫₀^π √(u²) dx = ∫₀^π |1 - 2sin(x/2)| dx</p><p><strong>Step 3:</strong> Determine sign changes on [0,π].</p><p>1 - 2sin(x/2) = 0 when sin(x/2) = 1/2, i.e., x/2 = π/6, so x = π/3</p><p>• For x ∈ [0, π/3]: sin(x/2) ≤ 1/2, so 1 - 2sin(x/2) ≥ 0</p><p>• For x ∈ [π/3, π]: sin(x/2) ≥ 1/2, so 1 - 2sin(x/2) ≤ 0</p><p><strong>Step 4:</strong> Split and integrate.</p><p>∫₀^(π/3) (1 - 2sin(x/2)) dx + ∫_(π/3)^π (2sin(x/2) - 1) dx</p><p>= [x + 4cos(x/2)]₀^(π/3) + [-4cos(x/2) - x]_(π/3)^π</p><p>= [π/3 + 4cos(π/6) - 0 - 4cos(0)] + [-4cos(π/2) - π - (-4cos(π/6) - π/3)]</p><p>= [π/3 + 4(√3/2) - 4] + [0 - π + 4(√3/2) + π/3]</p><p>= π/3 + 2√3 - 4 + π/3 + 2√3 - π</p><p>= 2π/3 - π + 4√3 - 4 = <strong>4√3 - 4 - π/3</strong></p><p>∴ Answer: A</p>
Correct Answer: A