Why can't you integrate x^x ie e ^ (x ln x) ?

Not sure of this but I guess any poly of magnitude above 4 is generally non-reducible if it forms part of a Galois group.

But I'd like a laymens explanation if it were possible

From SamuelPirme of advanced physics forums...

http://www.advancedphysics.org/viewthread.php?tid=1875&page=2

About Galois theory, it would take a long time to explain it and it has some prerequisites. In brief, it basically says that if you take a general quintic equation (so a polynomial of degree 5, or even one of higher degree), then the Galois group associated with it is not solvable (in general). (There's a technical meaning as to what this means in Group Theory.) For degree 4 polynomials (or less) the group is solvable, and this boils down to saying that you can in fact write down the solution in "analytic" form, just as you would the quadratic formula. (The groups under consideration are automorphism groups of some field extension that leave invariant a subfield, usually Q, the rational numbers.) ... Of course, if you are interested you can grab a Field Theory book and read it (or take a course).

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Still if you bound x ln x closely with an integral that can be integrated you might get a very close approximation!

E.g. x < x ln x < x^2, and its easy to integrate e^x or e^x^2 etc...

I feel smarter today - they are Liouville irreducibles http://mathworld.wolfram.com/Integral.html (1/2 way down)

Functions that can't be written as a finite series of elementary terms and opetations e.g like powers, or +/-/* etc...

They are integrable but the integrals are not expressed with a combination of a finite number of elementary functions.

Elliptic integrals are famous related with this question. The length of the arc of an ellipse is well defined and has a definite value, but it cannot be expressed with elementary functions.

Yep, bingo to a fellow mathematician if you follow that. I have pondered that for a long, long time. It took me google and a few hints to come across an area of maths I'd never explored before!