<p>∫₀⁴ ((y² + 4y + 5)sin(y−2))/(2y² + 8y + 1)dy is equal to</p>
Step-by-Step Solution
Key Concept: Recognizing symmetry properties of integrands and using substitution to transform the integral into an odd function over a symmetric interval.
<p><strong>Step 1:</strong> Complete the square in the numerator: y² + 4y + 5 = (y + 2)² + 1.</p><p><strong>Step 2:</strong> Complete the square in the denominator: 2y² + 8y + 1 = 2(y + 2)² − 7.</p><p><strong>Step 3:</strong> Make the substitution u = y − 2, so y = u + 2, and the integral becomes symmetric around u = 0 over [−2, 2].</p><p><strong>Step 4:</strong> Observe that the numerator contains sin(u), which is odd, while the overall integrand becomes odd after the substitution.</p><p><strong>Step 5:</strong> Since we're integrating an odd function over a symmetric interval, the result is 0.</p><p>∴ Answer is (A).</p>
Correct Answer: A