Suppose, I want to calculate $$I = \int\limits_{0}^1 (-1)^x dx.$$
Indeed, I'm not that good at integration, so I try to express $-1$ in some "simpler" way, like $$-1 = e^{i \cdot \pi}.$$
Then, integrating is pretty technical and I get $$I = \frac{2i}{\pi}.$$
But, otherwise I could pick just any other representation of $-1$, like $$-1 = e^{i \cdot (\pi + 2n\pi)}, \qquad n \in \mathbb{Z},$$ and the result would change accordingly to $$I = \frac{2i}{(2n+1)\pi}.$$
Now, my question is: does this mean that the integral is undefined, or is it defined but only when we pick the specific branch of $(-1)^z$ and what are those branches?