Step 1 :First, we set up the double integral: \(I = \int_{0}^{1} \int_{0}^{2} \frac{1+x^2}{1+y^2} dx dy\)
Step 2 :Next, we integrate with respect to x: \(I = \int_{0}^{1} \left[ x + \frac{x^3}{3} \right]_0^2 dy\)
Step 3 :Plug in the limits of integration for x: \(I = \int_{0}^{1} (2 + \frac{8}{3}) dy\)
Step 4 :Simplify the expression inside the integral: \(I = \int_{0}^{1} \frac{14}{3} dy\)
Step 5 :Integrate with respect to y: \(I = \left[ \frac{14}{3}y \right]_0^1\)
Step 6 :Plug in the limits of integration for y: \(I = \frac{14}{3}(1) - \frac{14}{3}(0)\)
Step 7 :Simplify the expression: \(I = \frac{14}{3}\)
Step 8 :\(\boxed{\frac{14}{3}}\) is the final answer.