Answer
$${e^2} - \sqrt e $$
Work Step by Step
$$\eqalign{
& \int_{1/2}^2 {\frac{{{e^{1/x}}}}{{{x^2}}}} dx \cr
& {\text{Integrate using the substitution method}} \cr
& u = \frac{1}{x},{\text{ }}du = - \frac{1}{{{x^2}}}dx,{\text{ }}dx = - {x^2}du \cr
& {\text{The new limits of integration are}} \cr
& x = 2 \to u = 1/2 \cr
& x = 1/2 \to u = 2 \cr
& {\text{Substitute and integrate}} \cr
& \int_{1/2}^2 {\frac{{{e^{1/x}}}}{{{x^2}}}} dx = \int_2^{1/2} {\frac{{{e^u}}}{{{x^2}}}} \left( { - {x^2}} \right)du \cr
& = - \int_2^{1/2} {{e^u}} du \cr
& = - \left[ {{e^u}} \right]_2^{1/2} \cr
& = - \left( {{e^{1/2}} - {e^2}} \right) \cr
& = {e^2} - \sqrt e \cr} $$
