No it's not, but the notation: ((2xe)^-x)^2 simplyfies to (2xe)^(-2x) not (2xe)^(-x^2)
i.e. (x^n)^m = x^(nm) != x^(n^m)
I'm not sure but someone once told me that you cannot integrate x^x. Are you sure that the question is possible?
(The derivative of x^x I can do though
).