<p>Let <br/>
\[ I = \int_{2}^{3} \frac{x^2\, dx}{2x^2 - 10x + 25} \]
Find the value of \(I\).</p>
Step-by-Step Solution
Key Concept: Recognize that the denominator 2x² - 10x + 25 = 2(x - 5/2)² is a perfect square, and decompose the numerator x² strategically as a combination of the denominator and its derivative to enable substitution.
<p><strong>Step 1: Identify the denominator structure</strong></p><p>2x² - 10x + 25 = 2(x² - 5x + 25/2) = 2(x - 5/2)²</p><p><strong>Step 2: Decompose the numerator</strong></p><p>Write x² = a(2x² - 10x + 25) + b(4x - 10) + c, where 4x - 10 is the derivative of the denominator.</p><p>Expanding: x² = 2a·x² + (−10a + 4b)x + (25a − 10b + c)</p><p>Comparing coefficients:</p><ul><li>x²: 2a = 1 ⟹ a = 1/2</li><li>x¹: −10a + 4b = 0 ⟹ −5 + 4b = 0 ⟹ b = 5/4</li><li>x⁰: 25a − 10b + c = 0 ⟹ 25/2 − 25/2 + c = 0 ⟹ c = 0</li></ul><p><strong>Step 3: Rewrite the integral</strong></p><p>I = ∫₂³ [1/2 + (5/4)(4x - 10)/(2x² - 10x + 25)] dx</p><p>I = [x/2]₂³ + (5/4)∫₂³ (4x - 10)/(2x² - 10x + 25) dx</p><p><strong>Step 4: Evaluate using substitution</strong></p><p>For the second integral, let u = 2x² - 10x + 25, then du = (4x - 10)dx</p><p>∫ (4x - 10)/(2x² - 10x + 25) dx = ln|2x² - 10x + 25|</p><p>At x = 3: 2(9) − 30 + 25 = 13</p><p>At x = 2: 2(4) − 20 + 25 = 13</p><p><strong>Step 5: Combine results</strong></p><p>I = [3/2 − 2/2] + (5/4)[ln(13) − ln(13)]</p><p>I = 1/2 + 0</p><p>∴ Answer: <strong>0.5</strong></p>
Correct Answer: 0.5