<p>Evaluate the integral: \[\int_{1/3}^{1} \frac{\pi \cos\left(\dfrac{\pi}{2}x\right)}{2\sin^2\left(\dfrac{\pi}{2}x\right)}\, dx\]</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand has the form f'(x)/f(x) structure after factoring out constants. The numerator is related to the derivative of the denominator, making substitution u = sin(πx/2) optimal.
<p><strong>Step 1:</strong> Let u = sin(πx/2), then du = (π/2)cos(πx/2)dx, so cos(πx/2)dx = (2/π)du</p><p><strong>Step 2:</strong> Rewrite the integral: ∫ (π/2u²)·(2/π)du = ∫ (1/u²)du</p><p><strong>Step 3:</strong> Change bounds: When x = 1/3, u = sin(π/6) = 1/2; when x = 1, u = sin(π/2) = 1</p><p><strong>Step 4:</strong> Evaluate: ∫₁/₂¹ u⁻² du = [-1/u]₁/₂¹ = -1/1 - (-1/(1/2)) = -1 + 2 = 1</p><p>∴ Answer: A</p>
Correct Answer: A