Is it possible to compute the double integral of 1/(1 + x^2 * y^2), where x varies between 0 and 1, and y varies between o and 1, and possibly express the answer in terms of special functions or numbers occuring in mathematics?

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.

Learn how to compute the double integral of 1/(1 + x^2 y^2) where x varies from 0 to 1 and y varies from 0 to 1. The solution involves advanced techniques including the use of special functions and mathematical constants. Explore detailed steps and insights into evaluating this integral.

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