Definite Integration
Estimation of Definite Integrals
Grade 12
Question:
<p>If \(f(x) = \displaystyle\int_2^x \frac{dt}{1+t^4}\), then:</p>
<p>(a) \(f(3) < \dfrac{1}{17}\)</p>
<p>(b) \(f(3) > \dfrac{1}{17}\)</p>
<p>(c) \(f(3) = \dfrac{1}{17}\)</p>
<p>(d) \(f(3) > 1\)</p>
Step-by-Step Solution
Key Concept: Recognize that f(x) is defined by the Fundamental Theorem of Calculus, so f'(x) equals the integrand evaluated at x. The derivative of a definite integral with variable upper limit directly gives the integrand at that limit.
<p><strong>Step 1:</strong> Recognize that f(x) = ∫₂ˣ dt/(1+t⁴) is defined as a function of the upper limit x.</p><p><strong>Step 2:</strong> Apply the Fundamental Theorem of Calculus: d/dx[∫₂ˣ f(t)dt] = f(x).</p><p><strong>Step 3:</strong> Therefore, f'(x) = 1/(1+x⁴).</p><p><strong>Step 4:</strong> Note that f(2) = 0 (integral from 2 to 2 is zero) and f is increasing on [2,∞) since f'(x) > 0 for all x.</p><p>∴ Answer: A</p>
Correct Answer: A