The double integral in question is:

$I = \int_{0}^{1} \int_{0}^{1} \frac{1}{1 + x^2 y^2} \, dx \, dy$

This integral cannot be expressed in terms of elementary functions, but it can be evaluated and expressed in terms of special functions and known mathematical constants.

Approach to Solution:

1. Interchange of Integrals: We can interchange the order of integration since the limits are independent:

   $I = \int_{0}^{1} \int_{0}^{1} \frac{1}{1 + x^2 y^2} \, dx \, dy = \int_{0}^{1} \left(\int_{0}^{1} \frac{1}{1 + x^2 y^2} \, dx\right) dy$

2. Integration with Respect to $x$: The inner integral with respect to $x$ can be computed by recognizing that it resembles the arctangent integral:

   $\int_{0}^{1} \frac{1}{1 + x^2 y^2} \, dx$ This can be simplified using the substitution $u = x y$, leading to:

   $\int_{0}^{1} \frac{1}{1 + x^2 y^2} \, dx = \frac{1}{y} \int_{0}^{y} \frac{1}{1 + u^2} \, du = \frac{1}{y} \arctan(y)$

3. Integration with Respect to $y$: Now substitute this result back into the outer integral:

   $I = \int_{0}^{1} \frac{\arctan(y)}{y} \, dy$

   This integral is a known special integral and can be evaluated as:

   $I = \frac{\pi}{4} \log(2)$

Conclusion:

The double integral $\int_{0}^{1} \int_{0}^{1} \frac{1}{1 + x^2 y^2} \, dx \, dy$ can be expressed in terms of the mathematical constants $\pi$ and $\log(2)$, with the final result being:

$I = \frac{\pi}{4} \log(2)$

Would you like more details on the steps, or do you have any further questions?

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Here are 8 related questions you might be interested in:

1. How is the arctangent function involved in other integrals?
2. Can the method used here be applied to similar double integrals?
3. What are some applications of the integral result $\frac{\pi}{4} \log(2)$?
4. How does this integral relate to other known integrals in mathematics?
5. Are there other ways to compute the integral $\int_{0}^{1} \frac{\arctan(y)}{y} \, dy$?
6. How does this integral behave if the limits of integration are changed?
7. What role do special constants like $\pi$ and $\log(2)$ play in mathematics?
8. Can numerical methods be used to verify this result?

Tip: When evaluating double integrals, consider if the integral can be simplified by changing the order of integration or by using substitutions that relate the integral to known special functions.