show that $f$ is not integrable on $[0, 1]$

Question

Define $f:[0,1] \rightarrow \mathbb{R}$ by

$f(x):= \begin{cases} e^{x}, \ x \in \mathbb{Q}\\ e^{-x}, \ x\in \mathbb{Q}^{c}\\ \end{cases}$

show that $f$ is not integrable on $[0,1]$.

I just started on analysis and though I'm somewhat familiar on showing a function is integrable, I'm find myself kind of stuck when trying to prove the opposite. I wanted to make use of the negation of the definition of darboux integrable, but do not know how to proceed. Any help or insights is deeply appreciated.

Answer 1

Hint: argue via upper and lower Riemann sums, using the that $\mathbb{Q}$ is dense.

(The upper Riemann sum will be the integral over $e^x$, the lower the one for $e^{-x}$.)