Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
127. ∫ (ln x) / (x + x ln x) dx

Verified step by step guidance

1. Start by examining the integral: \(\int \frac{\ln x}{x + x \ln x} \, dx\). Notice that the denominator can be factored to simplify the expression.
2. Factor the denominator: \(x + x \ln x = x(1 + \ln x)\). Rewrite the integral as \(\int \frac{\ln x}{x(1 + \ln x)} \, dx\).
3. Consider a substitution to simplify the integral. Let \(u = 1 + \ln x\). Then, compute \(du\) in terms of \(dx\): since \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), we have \(du = \frac{1}{x} dx\).
4. Rewrite the integral in terms of \(u\): replace \(\ln x\) with \(u - 1\) and \(\frac{1}{x} dx\) with \(du\). The integral becomes \(\int \frac{u - 1}{u} \, du\).
5. Simplify the integrand: \(\frac{u - 1}{u} = 1 - \frac{1}{u}\). Now, split the integral into two simpler integrals: \(\int 1 \, du - \int \frac{1}{u} \, du\). These can be integrated using basic rules.

Key Concepts

Integration by Substitution

Integration by substitution involves changing variables to simplify an integral. By identifying a part of the integrand as a function and its derivative elsewhere in the integral, you can replace it with a single variable, making the integral easier to solve.

Properties of Logarithmic Functions

Understanding logarithmic functions, especially ln(x), is crucial since they often appear in integrals. Knowing how to manipulate ln(x), such as recognizing its derivative (1/x) and how it behaves inside expressions, helps in simplifying and integrating complex expressions.

Algebraic Simplification of the Integrand

Before integrating, simplifying the integrand by factoring or canceling terms can make the integral more manageable. In this problem, rewriting the denominator and numerator to reveal common factors or simpler expressions is key to choosing the right integration technique.