<p><span>∫₀⁴ [(y² - 4y + 5)sin(y - 2)]/(2y² - 8y + 11)dy</span> is equal to:</p>
Step-by-Step Solution
Key Concept: Recognize that the integrand is an odd function about y = 2 by substituting u = y - 2, transforming the integral to show the numerator is odd while the denominator is even about the new origin.
<p><strong>Step 1:</strong> Let u = y - 2, so y = u + 2 and dy = du. When y = 0, u = -2; when y = 4, u = 2.</p><p>The integral becomes: ∫₋₂² [(u + 2)² - 4(u + 2) + 5]sin(u) / [2(u + 2)² - 8(u + 2) + 11] du</p><p><strong>Step 2:</strong> Simplify the numerator: (u + 2)² - 4(u + 2) + 5 = u² + 4u + 4 - 4u - 8 + 5 = u² + 1</p><p><strong>Step 3:</strong> Simplify the denominator: 2(u + 2)² - 8(u + 2) + 11 = 2(u² + 4u + 4) - 8u - 16 + 11 = 2u² + 8u + 8 - 8u - 16 + 11 = 2u² + 3</p><p><strong>Step 4:</strong> The integral becomes: ∫₋₂² [(u² + 1)sin(u)] / (2u² + 3) du</p><p><strong>Step 5:</strong> Note that the denominator (2u² + 3) is an even function: f(-u) = 2(-u)² + 3 = 2u² + 3 = f(u)</p><p><strong>Step 6:</strong> The numerator (u² + 1)sin(u) is a product of an even function (u² + 1) and an odd function sin(u), which makes it an odd function: g(-u) = ((-u)² + 1)sin(-u) = (u² + 1)(-sin(u)) = -g(u)</p><p><strong>Step 7:</strong> Therefore, the integrand [(u² + 1)sin(u)] / (2u² + 3) is odd (odd/even = odd).</p><p><strong>Step 8:</strong> By the property of odd functions, ∫₋₂² [odd function] du = 0</p><p><strong>∴ Answer: a</strong></p>
Correct Answer: a