[tex]\displaystyle \int_{0}^2 \sqrt[n]{x^n+(2-x)^n}\, dx =\\ \\ = \int_{0}^1\sqrt[n]{x^n+(2-x)^n}+ \int_{1}^2\sqrt[n]{x^n+(2-x)^n}= I_1+I_2\\ \\\\ I_1:\quad 0\leq x\leq 1 \Rightarrow x^n \to 0,\quad \text{cand }n\to \infty \\\\ \Rightarrow I_{1} = \int_{0}^1\sqrt[n]{(2-x)^n}\, dx= \int_{0}^1 (2-x)\, dx\\ \\ I_2:\quad 1\leq x\leq 2\Rightarrow (2-x)^n \to 0,\quad \text{cand }n\to \infty \\ \\ \Rightarrow I_2 = \int_{1}^2 \sqrt[n]{x^n}\, dx = \int_{1}^2 x\, dx[/tex]
[tex]\displaystyle \Rightarrow \lim\limits_{n\to \infty} \int_{0}^2 \sqrt[n]{x^n+(2-x)^n}\, dx = \int_{0}^1 (2-x)\, dx +\int_{1}^2 x\, dx = \\ \\ =\Big(2x-\dfrac{x^2}{2}\Big)\Bigg|_{0}^1+\dfrac{x^2}{2}\Bigg|_{1}^2 = 2-\dfrac{1}{2} +2 - \dfrac{1}{2} = \boxed{3}[/tex]
