Upper and lower integrals, piecewise function [duplicate]

Question: Let $f: [a,b] \to \mathbb{R}$ be a function such that$$f(x) = \begin{cases} x & x\in\Bbb Q \\ x+1 & x\not\in\Bbb Q\end{cases}.$$Compute the upper and lower integral of f.

So I'm guessing that since, given an arbitrary partition $P$ of $[a,b]$, each interval $[t_{i-1}, t_i]$ contains rational and irrational numbers, the upper integral would be $\int_a^b(x+1)dx = \frac{b^2-a^2}{2} + b - a$ and the lower integral $\int_a^bxdx = \frac{b^2-a^2}{2}$, and that would imply that f is not integrable...

Is this correct?