Determine whether the integral is convergent or divergent. If it is convergent, el Round to two decimal places, if necessary. \( \int_{2}^{\infty} \frac{\ln (x)}{x} d x \) Convergent: Divergent

To determine whether the integral \( \int_{2}^{\infty} \frac{\ln (x)}{x} d x \) is convergent or divergent, we can evaluate it. Let's change the variable \( u = \ln(x) \), which gives \( du = \frac{1}{x}dx \). The limits change accordingly: when \( x = 2 \), \( u = \ln(2) \) and as \( x \) approaches infinity, \( u \) approaches infinity. This transforms the integral to \( \int_{\ln(2)}^{\infty} u \, du \). Evaluating this integral, we find \( \int u \, du = \frac{u^2}{2} \), which diverges to infinity as \( u \) approaches infinity. Thus, the integral diverges. Divergent. Additionally, you can think of this integral in terms of behavior as \( x \) gets very large. The function \( \frac{\ln(x)}{x} \) grows slower than any polynomial function, but still, since the logarithm continues to rise indefinitely, it causes the area under the curve to not settle down to a finite value as we stretch to infinity. This reinforces the idea that we're not getting a nice, cozy convergence — it's a wild ride to infinity instead!