$f(x) = \begin{cases} \frac{1}{x^2} & \text{for x rational} \\ -\frac{1}{x^2} & \text{for x irrational} \\ \end{cases}$
Is it integrable from $1$ to $\infty$?
A book says it is, citing the reason that $|f(x)|$ is integrable and any absolutely integrable function is integrable.
But I am having doubt because I feel that for any partition P Upper Darboux Sum and Lower Darboux Sum are negative of each other, and they are non-zero. So, they will not converge to the same limit.
Any help in this regard is appreciated.
It is not true that Riemann integrability of $|f|$ implies that of $f$. Perhaps the book is talking about Lebesgue integral in which case this implications holds for all measurable functions $f$. This function is not Riemann integrable.
See https://en.wikipedia.org/wiki/Riemann_integral