Motivation part. I am currently thinking about extending real numbers with divergent integrals. I introduced some rules, for instance, for integrals of monomials, $\int_0^\infty x^n dx=\frac{\left(\tau +\frac{1}{2}\right)^{n+2}-\left(\tau -\frac{1}{2}\right)^{n+2}}{(n+1)(n+2)}$ where $\tau$ is th...