<p>Let a function <i>h(x)</i> be defined as <i>h(x)</i> = 0, for all <i>x</i> ≤ 0. Also <i>∫h(x).f(x)dx = f(0)</i> for every function <i>f(x)</i>. Then the value of the definite integral <i>∫₋₁⁰ h'(x).sin x dx</i> is</p>
Step-by-Step Solution
Key Concept: Understanding the properties of the Dirac delta function and recognizing that h'(x) = 0 on the interval [−1, 0].
<p><strong>Analysis:</strong> Using the given property that ∫h(x).f(x)dx = f(0) for every function f(x), we can identify h(x) as the Dirac delta function δ(x). For x ≤ 0, h(x) = 0, so h'(x) = 0 almost everywhere on [−1, 0].</p><p>Therefore, ∫₋₁⁰ h'(x).sin x dx = 0</p><p>∴ Answer is (A).</p>
Correct Answer: A