Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p><strong>Paragraph for Question nos. 642 and 643</strong><br>Let \(f\) be a continuous function such that \(g(x) = \displaystyle\int_{-1}^{1} f(t)|x-t|\, dt\) where \(x \in (-1,1)\).</p><p>The value of \(\displaystyle\int_{0}^{1} \dfrac{g''(x)}{f(x)}\, dx\) is equal to:</p>
<p>(a) \(0\)</p>
<p>(b) \(1\)</p>
<p>(c) \(2\)</p>
<p>(d) \(3\)</p>
Step-by-Step Solution
Key Concept: The function g(x) involves an absolute value in the integrand. By splitting the integral at t=x and removing the absolute value, we get g(x) = ∫₋₁ˣ f(t)(x-t)dt + ∫ₓ¹ f(t)(t-x)dt, which allows us to find g''(x) by differentiation.
<p><strong>Step 1:</strong> Split the integral at t = x to handle the absolute value:</p><p>g(x) = ∫₋₁ˣ f(t)(x-t)dt + ∫ₓ¹ f(t)(t-x)dt</p><p><strong>Step 2:</strong> Differentiate g(x) using Leibniz rule:</p><p>g'(x) = ∫₋₁ˣ f(t)dt + f(x)·0 - ∫ₓ¹ f(t)dt - f(x)·0</p><p>g'(x) = ∫₋₁ˣ f(t)dt - ∫ₓ¹ f(t)dt</p><p><strong>Step 3:</strong> Differentiate g'(x) to find g''(x):</p><p>g''(x) = f(x) - (-f(x)) = 2f(x)</p><p><strong>Step 4:</strong> Evaluate the integral:</p><p>∫₀¹ g''(x)/f(x) dx = ∫₀¹ 2f(x)/f(x) dx = ∫₀¹ 2 dx = 2</p><p>∴ Answer: C</p>
Correct Answer: C