$I=\int\_{1}^{2}\frac{e^x+e^{4/x}}{x}dx$

For $ x \in (1,2), \frac{e^x+e^{4/x}}{x}>e^x$

$\Rightarrow I=\int\_{1}^{2}\frac{e^x+e^{4/x}}{x}dx$ >$\int\_{1}^{2}e^xdx$

$\Rightarrow I>e^2-e$

And $I<e+e^4$ by considering a rectangle

Is there a better way we can approximate the integral more efficiently by assuming some function like $e^x$ which can be understood by a highschool student?