We know that some functions such as $x^x$ do not have anti-derivatives. But, just because such functions do not have anti derivatives, does that mean we simply cannot find the area under the graph? Or are there any methods in calculus that can be used to find the area under the curve of such functions which cannot be integrated.
Edit: In this context, let us consider a non integrable function as one which does not have a closed form of an anti derivative.