Definite Integration
Properties of Definite Integrals
Grade 12
Question:
<p>If \(f(x) = \displaystyle\int_2^x \frac{dt}{1+t^4}\), then:</p>
<p>\(f(3) < \dfrac{1}{17}\)</p>
<p>\(f(3) > \dfrac{1}{17}\)</p>
<p>\(f(3) = \dfrac{1}{17}\)</p>
<p>\(f(3) > 1\)</p>
Step-by-Step Solution
Key Concept: Use Leibniz rule for differentiation under the integral sign: if F(x) = ∫[a to x] f(t)dt, then F'(x) = f(x). Apply this directly to find f'(x) = 1/(1+x⁴), then use properties of definite integrals to relate f(x) to other bounds.
<p><strong>Step 1:</strong> Apply Leibniz rule for differentiation under the integral sign.</p><p>Given: f(x) = ∫₂ˣ dt/(1+t⁴)</p><p><strong>Step 2:</strong> By fundamental theorem of calculus, differentiate both sides with respect to x:</p><p>f'(x) = d/dx[∫₂ˣ dt/(1+t⁴)] = 1/(1+x⁴)</p><p><strong>Step 3:</strong> Note boundary conditions: f(2) = ∫₂² dt/(1+t⁴) = 0</p><p><strong>Step 4:</strong> The function f(x) is strictly increasing for all x since f'(x) = 1/(1+x⁴) > 0 for all real x.</p><p><strong>Step 5:</strong> For x > 2: f(x) > 0, and for x < 2: f(x) < 0. Also, f is continuous and differentiable everywhere.</p><p>∴ Answer: B</p>
Correct Answer: B