<p>We have <span>\(F(x) + F\!\left(x + \dfrac{1}{2}\right) = 3\)</span>. Find the value of <span>\(\int_0^2 F(x)\, dx\)</span>.</p>
Step-by-Step Solution
Key Concept: Use the functional equation F(x) + F(x + 1/2) = 3 to create a relationship between integrals over complementary intervals. Substitute x with x + 1/2 to get a second equation, then add strategically.
<p><strong>Step 1:</strong> Write the given functional equation:</p><p>F(x) + F(x + 1/2) = 3 ... (1)</p><p><strong>Step 2:</strong> Replace x with (x + 1/2) in equation (1):</p><p>F(x + 1/2) + F(x + 1) = 3 ... (2)</p><p><strong>Step 3:</strong> Integrate equation (1) from 0 to 2:</p><p>∫₀² F(x) dx + ∫₀² F(x + 1/2) dx = ∫₀² 3 dx = 6</p><p><strong>Step 4:</strong> In the second integral, substitute u = x + 1/2, so du = dx. When x = 0, u = 1/2; when x = 2, u = 5/2:</p><p>∫₀² F(x) dx + ∫₁/₂^(5/2) F(u) du = 6</p><p><strong>Step 5:</strong> Split the second integral:</p><p>∫₀² F(x) dx + ∫₁/₂² F(x) dx + ∫₂^(5/2) F(x) dx = 6</p><p><strong>Step 6:</strong> Rearrange:</p><p>∫₀^(1/2) F(x) dx + 2∫₁/₂² F(x) dx + ∫₂^(5/2) F(x) dx = 6</p><p><strong>Step 7:</strong> By symmetry of the functional equation over the interval [0,2], and noting that the relationship holds uniformly, we can determine that ∫₀² F(x) dx splits equally. From the functional equation structure:</p><p>∫₀² F(x) dx + ∫₀² F(x + 1/2) dx = 6 yields 2∫₀² F(x) dx = 6</p><p>∴ ∫₀² F(x) dx = <strong>3</strong></p>
Correct Answer: C