Double integral of 1/(1 + x^2 *y^2) with respect to x and y

To solve the double integral of \(\frac{1}{1 + x^2 y^2}\) with respect to \(x\) and \(y\), we can express it as:

\[
I = \iint \frac{1}{1 + x^2 y^2} \, dx \, dy
\]

### Step 1: Consider the order of integration
Let's assume the region of integration is a rectangle for simplicity, say \(x \in [a,b]\) and \(y \in [c,d]\). The double integral can then be written as:

\[
I = \int_c^d \left( \int_a^b \frac{1}{1 + x^2 y^2} \, dx \right) dy
\]

### Step 2: Integrate with respect to \(x\)
First, we need to compute the inner integral:

\[
\int_a^b \frac{1}{1 + x^2 y^2} \, dx
\]

Let's use the substitution \( u = x \sqrt{y^2} \), or \( u = xy \), then \( du = y \, dx \) and \( dx = \frac{du}{y} \):

\[
\int_a^b \frac{1}{1 + x^2 y^2} \, dx = \frac{1}{y} \int_{ay}^{by} \frac{1}{1 + u^2} \, du
\]

The integral \(\int \frac{1}{1 + u^2} \, du\) is well-known and equals \(\arctan(u)\):

\[
\frac{1}{y} \left[ \arctan(u) \right]_{ay}^{by} = \frac{1}{y} \left( \arctan(by) - \arctan(ay) \right)
\]

### Step 3: Integrate with respect to \(y\)
Now, the original double integral becomes:

\[
I = \int_c^d \frac{1}{y} \left( \arctan(by) - \arctan(ay) \right) dy
\]

This integral needs to be evaluated based on the specific limits \(c\) and \(d\). If specific limits are provided, we can proceed with evaluating this integral directly. If the limits are infinite, or involve more specific functions, the evaluation could require more sophisticated techniques.

### Special Case: Limits from 0 to ∞
If the limits for both \(x\) and \(y\) are from \(0\) to \(\infty\), the integral simplifies further and may be approached using techniques like Fubini's theorem or polar coordinates.

Would you like to specify the limits of integration, or should we explore further with general methods? 

#### Related Questions:
1. What substitution method can be applied if the limits of integration are infinite?
2. How would the problem change if the limits were \([-1, 1]\) instead of general limits?
3. What are the typical applications of such integrals in physics or engineering?
4. How does this integral relate to special functions like Beta or Gamma functions?
5. Can this integral be solved using polar coordinates? If so, how?
6. What are the implications of the integral if \(x\) and \(y\) represent real-world quantities like time or distance?
7. How does changing the function in the denominator affect the difficulty of the integral?
8. What numerical methods could be used to approximate this integral if an exact solution is complex?

**Tip:** Always check if a problem can be simplified by changing the order of integration or applying symmetry properties.

Explore the solution to the double integral of 1/(1 + x^2 * y^2) using integration techniques and substitution method. Understand the application of Fubini's theorem and the arctangent function in solving this integral. Suitable for advanced undergraduate and higher levels.

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