This type integral can't be found a symbolic solution - closed-form solution. With numerics is easy:
f[n_?NumericQ] :=
NIntegrate[Sqrt[n^2 - x^2]/(2 + x^(-x)), {x, 0, n}]; Table[
f[n]/n^2 /. n -> 10^k, {k, 1, 6}]
(*{0.359439, 0.389349, 0.392364, 0.392699, 0.392699, 0.392699}*)
Looks like limit have a value: 0.392699...
EDITED:
We can use:
AsymptoticLessEqual[x^-x, 1/(x^2 + 1), x -> \[Infinity]]
(*True*)
then:
Limit[Integrate[
Sqrt[n^2 - x^2]/(2 + (1/(x^2 + 1))) // Factor, {x, 0, n},
Assumptions -> {n > 0}]/n^2, n -> Infinity]
(*\[Pi]/8*)
\[Pi]/8 // N
(*0.392699*)