Sorry, you'll have to define what you mean FAR better than that.
Your examples with improper integrals are bad examples because "integrable" is defined on closed bounded intervals.it all depends on how you define integrable
Also, I believe is another integral that can't be solved...
That integral came up in my Differential Equations class when solving (this was the correct equation). He gave us , and we wasted 45 minutes trying to figure that one out... :'(