Integral $\int_{-1}^{1} \frac{\mathrm dx}{x}$

Question

Consider the following integration $$\int_{-1}^1\frac{\mathrm dx}{x}$$ B thinks this expression should be written as $$\int_{-1}^0\frac{\mathrm dx}{x}+\int_0^1\frac{\mathrm dx}{x}$$ which is not convergent, but A thinks it is zero because it is an odd function with the symmetric region so the integration is zero. B thinks that the integration doesn't converge, so we are not allowed to use the properties of integration.

I think B is correct. Right?

Answer 1

Student B is correct in the sense that traditionally, this integral cannot be evaluated. However, the Cauchy principal value of this integral exists (the 'intuitive' way to extend integrals into areas where they do not converge) and is indeed 0.