If $|f|$ can be integrated using Riemann integration, does that mean that $f$ can be integrated using Riemann integration?
Also
If $f^{2}$ can be integrated using Riemann integration, does that mean that $f$ can be integrated using Riemann integration?
Are there any examples?
On
Pick the function: $f(x) = \begin{cases} 1, & \text{if } x \in \mathbb{R \setminus Q} \\ -1, & \text{if } x \in \mathbb{Q} \end{cases} $
It is easy to see, this function is discontinuous everywhere and hence not Riemann integrable on any intervall, upper and lower sum differ.
However$ |f(x)| = 1 \ \forall \ x \in \mathbb{R} $ and integrable sufficing your first question.
Let $f(x)=1$ if $x$ is rational, and $-1$ if $x$ is irrational. The functions $|f|$ and $f^2$ are Riemann integrable on $[0,1]$, but $f$ is not.