Concept: Split integral at \(x=1\) where \(\ln x\) changes sign. Integrate \(\frac{\ln x}{x}\) using substitution.
Step 1: For \(x\in[1/e,1)\), \(\ln x<0\): \(|\ln x|/x = -\ln x/x\). For \(x\in[1,e^2]\), \(\ln x\ge0\): \(|\ln x|/x = \ln x/x\). Integral = \(-\int_{1/e}^1 \frac{\ln x}{x}dx + \int_1^{e^2} \frac{\ln x}{x}dx\).
Step 2: \(\int \frac{\ln x}{x}dx = \frac{(\ln x)^2}{2}\). Evaluate: \(-\left[0 - \frac{1}{2}\right] + \left[\frac{4}{2} - 0\right] = \frac12 + 2 = \frac52\).
Step 3: Write the final answer. \(\boxed{\frac52}\)