Definite Integration
Properties of definite integrals with even/odd functions
Grade 12
Question:
<p>If \(f(x)\) is an even function, then the value of \(\int_{-2}^{2} [x^2 f'(x) - x^3 f''(x) - \lambda] dx\), where \(\lambda\) is a constant, is</p>
<p>(a) 4\(\lambda\)</p>
<p>(b) \(f(2) - 4\lambda\)</p>
<p>(c) \(2f^2(2) + 4\lambda\)</p>
<p>(d) none of these</p>
Step-by-Step Solution
Key Concept: For an even function f(x), use symmetry properties: x²f'(x) is odd (even × odd = odd) and x³f''(x) is even (odd × even = odd), so their integrals over symmetric intervals have specific properties. Apply integration by parts strategically to leverage these symmetries.
<p><strong>Step 1: Analyze parity of each term</strong></p><p>Since f(x) is even: f(-x) = f(x), so f'(-x) = -f'(x) (f' is odd) and f''(-x) = f''(x) (f'' is even).</p><p><strong>Step 2: Determine parity of integrand components</strong></p><p>• x²f'(x): [even] × [odd] = odd function → ∫₋₂² x²f'(x)dx = 0</p><p>• x³f''(x): [odd] × [even] = odd function → ∫₋₂² x³f''(x)dx = 0</p><p><strong>Step 3: Evaluate the constant term</strong></p><p>∫₋₂² (-λ)dx = -λ[x]₋₂² = -λ(2-(-2)) = -4λ</p><p><strong>Step 4: Combine results</strong></p><p>∫₋₂² [x²f'(x) - x³f''(x) - λ]dx = 0 - 0 - 4λ = <strong>-4λ</strong></p><p>∴ Answer: D</p>
Correct Answer: D