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

Verified step by step guidance

1. 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 \(x^{2x}\).
2. Rewrite the integrand by expressing \(x^{2x}\) in terms of the exponential function: \(x^{2x} = e^{2x \ln x}\).
3. Differentiate \(x^{2x}\) with respect to \(x\) using the chain rule: \(\frac{d}{dx} x^{2x} = \frac{d}{dx} e^{2x \ln x} = e^{2x \ln x} \cdot \frac{d}{dx} (2x \ln x)\).
4. Calculate \(\frac{d}{dx} (2x \ln x)\) using the product rule: \(\frac{d}{dx} (2x \ln x) = 2 \ln x + 2\).
5. 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.

Key Concepts

Integration by Substitution

Integration by substitution is a technique used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This method is especially useful when the integral contains a function and its derivative.

Differentiation of Exponential Functions with Variable Exponents

Functions like x^(2x) involve variable exponents, which require logarithmic differentiation to handle. Understanding how to differentiate and integrate such functions is crucial, as they combine polynomial and exponential behaviors. Recognizing the derivative of the exponent helps in integration.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus links differentiation and integration, allowing evaluation of definite integrals using antiderivatives. After finding an antiderivative of the integrand, you compute its values at the upper and lower limits and subtract to find the integral's value.