Definite Integration
Properties of definite integrals
Grade None
Question:
<p>Given that \(g(x) = \int_0^x \cos 4t\,dt\), which of the following is true?</p>
<p>\(g(x + \pi) = g(x) + g(\pi)\)</p>
<p>\(g(x + \pi) = g(x) - g(\pi)\)</p>
<p>\(g(x + \pi) = g(x) \cdot g(\pi)\)</p>
<p>\(g(x + \pi) = g(x)/g(\pi)\)</p>
Step-by-Step Solution
Key Concept: Recognize that g(x) is defined by a definite integral with variable upper limit. Use the Fundamental Theorem of Calculus: if g(x) = ∫₀ˣ f(t)dt, then g'(x) = f(x). Here, g'(x) = cos(4x), and g(0) = 0 is the initial condition.
<p><strong>Step 1:</strong> Evaluate g(0) using the property of definite integrals: g(0) = ∫₀⁰ cos 4t dt = 0 (integral with equal limits = 0)</p><p><strong>Step 2:</strong> Find g'(x) using the Fundamental Theorem of Calculus: g'(x) = d/dx[∫₀ˣ cos 4t dt] = cos 4x</p><p><strong>Step 3:</strong> Evaluate the indefinite integral: g(x) = ∫ cos 4t dt = sin(4t)/4 + C. Since g(0) = 0, we have 0 = sin(0)/4 + C = 0 + C, so C = 0</p><p><strong>Step 4:</strong> Therefore, g(x) = sin(4x)/4 = (1/4)sin(4x). Verify: g(0) = 0 ✓ and g'(x) = cos(4x) ✓</p><p>∴ Answer: A</p>
Correct Answer: A