integration – How can you be sure that an integral does not exist? Not that it diverges, but that you cannot complete it?

Say you have the integral $$displaystyleint_1^infty{frac{1}{x^{1+frac{1}{x}}}};mathrm{d}x$$

This integral cannot be completed. Not that it goes to infinity, but it physically just cannot be completed. How can you realize this if you encounter it? How can you prove it?