Definite Integration
Differentiation under integral sign / Leibniz rule
Grade None
Question:
<p>Here, <span>\(F(x^2) = x^4 + x^5 = \int_0^{x^2} t\, f(t)\, dt\)</span>. Find <span>\(\sum_{r=1}^{12} f(r^2)\)</span>.</p>
<p>A) 210</p>
<p>B) 215</p>
<p>C) 219</p>
<p>D) 225</p>
Step-by-Step Solution
Key Concept: Differentiate both sides of F(x²) = ∫₀^(x²) t·f(t)dt with respect to x to find f(x²), then use the relationship 2x·F'(x²) = x²·f(x²) to extract f as a function of its argument.
<p><strong>Step 1:</strong> Differentiate both sides with respect to x:</p><p>Given: F(x²) = x⁴ + x⁵ = ∫₀^(x²) t·f(t)dt</p><p>Left side: d/dx[F(x²)] = F'(x²)·2x</p><p>Right side: d/dx[x⁴ + x⁵] = 4x³ + 5x⁴</p><p><strong>Step 2:</strong> Apply Leibniz rule to the integral:</p><p>d/dx[∫₀^(x²) t·f(t)dt] = x²·f(x²)·2x (by fundamental theorem)</p><p><strong>Step 3:</strong> Equate: 2x·F'(x²) = x²·f(x²)·2x = 4x³ + 5x⁴</p><p>Therefore: x²·f(x²) = 4x³ + 5x⁴</p><p>So: f(x²) = 4x + 5x²</p><p><strong>Step 4:</strong> Substitute u = x²: f(u) = 4√u + 5u</p><p><strong>Step 5:</strong> Calculate the sum:</p><p>∑ᵣ₌₁¹² f(r²) = ∑ᵣ₌₁¹² (4r + 5r²) = 4∑ᵣ₌₁¹² r + 5∑ᵣ₌₁¹² r²</p><p>= 4·(12·13/2) + 5·(12·13·25/6)</p><p>= 4·78 + 5·650 = 312 + 3250 = <strong>3562</strong></p><p>∴ Answer: C</p>
Correct Answer: C