Ch. 7 - Transcendental Functions

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Hass 15th Edition

Ch. 7 - Transcendental Functions

Problema 7.3.95

Capitolo 7, Problema 7.3.95

Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx

Guida verificata passo dopo passo

Recognize that the integrand is of the form \(x^{2x} (1 + \ln x)\), which suggests a function multiplied by its derivative or a derivative of a product involving \(x^{2x}\).

Rewrite the integrand by expressing \(x^{2x}\) in terms of the exponential function: \(x^{2x} = e^{2x \ln x}\).

Differentiate \(x^{2x}\) with respect to \(x\) using the chain rule: \(\frac{d}{dx} x^{2x} = \frac{d}{dx} e^{2x \ln x} = e^{2x \ln x} \cdot \frac{d}{dx} (2x \ln x)\).

Calculate \(\frac{d}{dx} (2x \ln x)\) using the product rule: \(\frac{d}{dx} (2x \ln x) = 2 \ln x + 2\).

Notice that the integrand \(x^{2x} (1 + \ln x)\) matches \(\frac{1}{2} \frac{d}{dx} x^{2x}\), so rewrite the integral accordingly and integrate by reversing the differentiation.

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