Gonna go on something tangential for a bit, but it'll make sense in a moment.
There are sums like 1+2+4+8+... which we simply leave undefined because they do not converge in R. However, convergence in R isn't the only "sensible" way to assign values. You could use convergence in Q_2 (2-adic numbers), where this sum converges to -1. And you can also use the formula 1 + x + x^2 + x^3 + ... = 1/(1-x) and analytic continuation (which also agrees with the result -1, but that's not the point). You could also justify it because it works out algebraically: S = 1+2+4+8+..., 2S = 2+4+8+16+..., so S-2S = 1, and -S = 1, and S = -1.
Integration is also a form of "sum", so even if the integral is undefined because the positive and negative parts aren't Lebesgue integrable, you could still define this integral in other ways, like algebraically: Split into -inf to 0 and 0 to inf, do a u-sub with u = -x, cancel identical integrals.
You can see here in this article it's discussed how you can reasonably define this integral, but also that approaching the singularity at two different rates causes it to not be zero:
https://en.wikipedia.org/wiki/Cauchy_principal_value#ExamplesThe problem with people in this thread is that they're treating math as a subject that only gives you absolute truths, rather than a subject where you have definitions and axioms and you use them to reach conclusions in a controlled way (and you may have contradictory conclusions based on different axioms/definitions).
You want to define it? Go ahead. You want to keep it undefined? Go ahead. You want to use infinity? Go ahead. You want to be a finitist? Go ahead. Just don't go around acting like yours is the only way.
