$\huge \begin{align} A &=\int_{2}^{e} \frac{\d x}{ 2x \pa{\ln x}^2 } \\ f(x) &= \int \frac{\d x}{2x(\ln x)^2}\\ \\ \let u &= \ln(x) \\ \let \d u &= x^{-1}\\ \\ f(x)&=\int \frac{1}{2} u^{-2} \d u \\ &= -\frac{1}{2} u^{-1} \d u \\ &= -\frac{1}{2\ln x}\\ \\ f(e)-f(2) &= -\frac{1}{2\ln e}- -\frac{1}{2\ln 2} \\ &= -\frac{1}{2}\pa{1- \frac{1}{\ln 2}} \\ \end{align}$