Past exam of the options II course of the University of Oxford / 2012 / D / D3 / c

For every real $x$, the numerator is at most $2e^{|x|}$, while the denominator is at least $e^{2|x|}$. ThereforeThe upper bound is integrable by part (a), so the comparison theorem shows that the function is integrable on $\mathbb{R}$.