Singularities in path integrals

So, I once watched a video (I think it was by Andrew Dotson) that I can’t find any more about techniques used to deal with singularities in path integrals. The presenter mentioned how the integral formula:

$$int x^{n} ; mathrm{d} x = frac{1}{n+1}x^{n+1} + C$$

cannot be used for $n=-1$. However, to deal with the singularity at $n=-1$, the presenter said to choose $C = – frac{1}{n+1}$, and taking the following limit:

$$lim_{n to -1} left( frac{1}{n+1}x^{n+1} – frac{1}{n+1} right)$$

You approach the natural log function as $n$ approaches $-1$. I think the presenter said later that a similar idea is used to deal with path integrals with singularities, but I forget what. How does this relate to getting rid of singularities, and what do these types of integrals with singularities represent in physics? I know that path integrals are used in Quantum Mechanics, so how must a particle behave that necessitates getting rid of a singularity in an integral?