Definite Integration
Properties of definite integrals
Grade 12
Question:
<p>\(\int_a^b f(x)dx = 2\int_a^b f(x)dx\) can hold if</p>
<p>(a) \(f(x)\) is even</p>
<p>(b) \(f(x)\) is odd</p>
<p>(c) \(f(x)\) is periodic</p>
<p>(d) \(f(2a-x) = f(x)\)</p>
<p>(e) \(f(2a-x) = f(x)\)</p>
Step-by-Step Solution
Key Concept: The equation ∫ₐᵇ f(x)dx = 2∫ₐᵇ f(x)dx simplifies to ∫ₐᵇ f(x)dx = 0, which occurs when a = b or when f(x) is an odd function integrated over a symmetric interval.
<p><strong>Step 1:</strong> Analyze the given equation: ∫ₐᵇ f(x)dx = 2∫ₐᵇ f(x)dx</p><p><strong>Step 2:</strong> Rearrange: ∫ₐᵇ f(x)dx - 2∫ₐᵇ f(x)dx = 0 → -∫ₐᵇ f(x)dx = 0 → ∫ₐᵇ f(x)dx = 0</p><p><strong>Step 3:</strong> This holds when: (i) <strong>a = b</strong> (upper limit equals lower limit, integral = 0 by definition), (ii) <strong>f(x) is odd and limits are symmetric</strong> like [-c, c] where negative and positive areas cancel, or (iii) <strong>f(x) = 0</strong> on [a,b]</p><p><strong>Step 4:</strong> The most general and non-trivial answer is <strong>a = b</strong>, as this guarantees both sides equal zero regardless of f(x).</p><p>∴ Answer: D (typically a = b, or the option stating equal limits)</p>
Correct Answer: D