Definite Integration Question - None

Question

If $ xf(x) = x + \int_1^x f(t)\,dt $, then find $ \sum_{k=1}^{10} f(e^k) $.

Accepted Answer

Step 1: Differentiate both sides of xf(x) = x + ∫₁ˣ f(t)dt with respect to x.

Left side: d/dx[xf(x)] = f(x) + xf'(x)

Right side: d/dx[x + ∫₁ˣ f(t)dt] = 1 + f(x)

Step 2: Equate derivatives: f(x) + xf'(x) = 1 + f(x)

This simplifies to: xf'(x) = 1, so f'(x) = 1/x

Step 3: Integrate: f(x) = ln(x) + C

Use initial condition from original equation: At x=1, f(1)·1 = 1 + ∫₁¹ f(t)dt = 1 + 0 = 1, so f(1) = 1

Therefore: 1 = ln(1) + C = 0 + C, giving C = 1

Thus: f(x) = ln(x) + 1

Step 4: Calculate f(eᵏ) = ln(eᵏ) + 1 = k + 1

Step 5: Compute the sum: ∑ₖ₌₁¹⁰ f(eᵏ) = ∑ₖ₌₁¹⁰ (k + 1) = ∑ₖ₌₁¹⁰ k + ∑ₖ₌₁¹⁰ 1

= [10(11)/2] + 10 = 55 + 10 = 65

∴ Answer: 65

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