Is this integral indeterminate?

It is indeterminate. But it is quite interesting to see why it is indeterminate. If you defined it to be the limit of the integral with extremes $a$ and $-a$ for $ a \to \infty$, then it would be zero. But that is not correct because, long story short, when you take a limit you don’t want the result to depend on how you arrived at that “limit point”. For example, consider it as a function of two variables:

$$ F(a,b)= \int_{a} ^b x dx $$

Then your integral would be the limit of this function as $a \to -\infty$ and $b \to \infty$. Check yourself that this limit doesn’t exist because it depends on how you approach to $\infty$ and $-\infty$.

Of course there are generalisations, or situations where you have to approximate things in a certain way (see the principal value distribution, for instance), but generally speaking that integral is not definite