Does $\int_{-1}^1 \frac 1 x dx$ equal zero? Does $\int_{-1}^1 \frac 1 x dx$ equal zero? My contention is that it should, however several sources have said that it does not. I would think that the two areas on the left and the right of $x=0$ should be inverses and thus cancel, even if their areas are each infinite.

They would cancel, but the integral doesn't exist, as 
$$\int_0^1 \frac{1}{x} \, \mathrm{d} x$$
doesn't exist. You need to calculate $\infty - \infty$ and that doesn't work. Regarding to the parametrisation you choose it won't be $0$.
