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.

Your intuition is formalized by the Cauchy Principal Value, which deals with this exact type of problem.
However, note that it does not exist in the Riemann integral, because both the positive and negative part deal with $\infty$, and $\infty - \infty$ is undefined.
