29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₁² (1 + ln x) x^x dx

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Identify the integral to solve: \(\int_1^2 (1 + \ln x) x^x \, dx\).

Recognize that the integrand contains the function \(x^x\), which can be rewritten using exponentials and logarithms as \(x^x = e^{x \ln x}\).

Differentiate \(x^x\) with respect to \(x\) to find its derivative: \(\frac{d}{dx} x^x = x^x (1 + \ln x)\), which matches the integrand's factor.

Use this observation to rewrite the integral as \(\int_1^2 \frac{d}{dx} x^x \, dx\), since the integrand is exactly the derivative of \(x^x\).

Apply the Fundamental Theorem of Calculus to evaluate the integral as \(x^x\) evaluated from 1 to 2, i.e., \(x^x \big|_1^2\).

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Integration of Functions Involving Logarithms

Understanding how to integrate functions that include logarithmic terms, such as ln(x), is essential. This often involves recognizing when to use integration by parts or substitution, especially when the integrand is a product of functions like (1 + ln x) and another function.

Handling Functions with Variable Exponents (x^x)

The function x^x is a variable base raised to a variable exponent, which is not a standard elementary function. To work with it, rewrite x^x as e^(x ln x), enabling differentiation or integration techniques involving exponentials and logarithms.

Integration by Parts

Integration by parts is a method used to integrate products of functions and is based on the product rule for differentiation. It is particularly useful when the integrand is a product like (1 + ln x) * x^x, allowing the integral to be broken down into simpler parts.

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.∫₁² (1 + ln x) x^x dx