For Exercises 127 and 128 find a function f satisfying each equation.
127. ∫₂ˣ √(f(t)) dt = x ln x

Guida verificata passo dopo passo

Recognize that the problem gives an integral equation: \(\int_{2}^{x} \sqrt{f(t)} \, dt = x \ln x\). Our goal is to find the function \(f(x)\) that satisfies this equation.

Apply the Fundamental Theorem of Calculus by differentiating both sides of the equation with respect to \(x\). This gives: \(\frac{d}{dx} \int_{2}^{x} \sqrt{f(t)} \, dt = \frac{d}{dx} (x \ln x)\).

By the Fundamental Theorem of Calculus, the derivative of the integral with variable upper limit is the integrand evaluated at \(x\): \(\sqrt{f(x)} = \frac{d}{dx} (x \ln x)\).

Compute the derivative on the right side using the product rule: \(\frac{d}{dx} (x \ln x) = \ln x + 1\).

Set \(\sqrt{f(x)} = \ln x + 1\) and solve for \(f(x)\) by squaring both sides: \(f(x) = (\ln x + 1)^2\).

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