[tex]\displaystyle I = \int_{0}^1 \dfrac{x^2+1}{x^4+1}\, dx = \int_{0}^1 \dfrac{x^{-2}(x^2+1)}{x^{-2}(x^4+1)}\, dx = \int_{0}^1 \dfrac{1+x^{-2}}{x^{2}+x^{-2}}\, dx= \\ \\ = \int_{0}^1 \dfrac{1+x^{-2}}{(x-x^{-1})^2+2}\, dx \\ \\ \\x-x^{-1} = t \Rightarrow (1+x^{-2})\, dx = dt \\ x\searrow 0 \Rightarrow t\to -\infty \\ x = 1 \Rightarrow t = 0[/tex]
[tex]\displaystyle I = \int_{-\infty}^{0}\dfrac{1}{t^2+2}\, dt = \dfrac{1}{\sqrt 2}\arctan \dfrac{t}{\sqrt 2}\Bigg|_{-\infty}^0 = 0-\dfrac{1}{\sqrt 2}\arctan(-\infty) = \\ \\ = -\dfrac{1}{\sqrt 2}\cdot \Big(-\dfrac{\pi}{2}\Big) = \boxed{\dfrac{\pi}{2\sqrt 2}}[/tex]
