Show given function is not integrable

Question

The function $f$ defined on $(0,1)$ by

$$ f(x)= \begin{cases} 0, \rm{if \ x \ is \ rational} \\ \lfloor \frac{1}{x} \rfloor, \rm{if \ x \ is \ irrational} \\ \end{cases} $$

Show that $\int_0^1 f(x) dx = \infty$

I'm not sure how to proceed in this. Just looking for some hints.

Edit: Since, irrationals are a set of infinite measure and $\lim_{x\rightarrow 0} f(x) = \infty$, can we directly say, the integral will be infinite?

Answer 1

Consider $g(x)=1/x-1$, $g(x)\leq f(x)$ almost everywhere and $\int_0^1g(x)=ln(x)-x]_0^1$.