The limit $\lim_\limits{n\to{\infty}}\int_0^{1}\sqrt[n]{x^n+(1-x)^n} dx$

Question

This is Problem 11941 in the American Mathematical Monthly.

What can we say about the limit $\lim_\limits{n\to{\infty}}\int_0^{1}\sqrt[n]{x^n+(1-x)^n} dx\ ?$

I think we need to use gamma and beta functions here. Again, the result may involve the Euler's constant.