(14) ∫(e² lnx / x) dx

Explanation:

Step1: Use substitution

Let $u = \ln x$, then $du=\frac{1}{x}dx$. When $x = e$, $u=\ln e = 1$; when $x=e^{2}$, $u=\ln e^{2}=2$.

Step2: Rewrite the integral

The integral $\int_{e}^{e^{2}}\frac{\ln x}{x}dx$ becomes $\int_{1}^{2}u du$.

Step3: Apply power - rule for integration

The power - rule for integration is $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$). For $\int_{1}^{2}u du=\left[\frac{u^{2}}{2}\right]_{1}^{2}$.

Step4: Evaluate the definite integral

$\frac{2^{2}}{2}-\frac{1^{2}}{2}=\frac{4}{2}-\frac{1}{2}=\frac{3}{2}$.

Answer:

$\frac{3}{2}$