To compute the double integral of $\frac{1}{1 + xy}$ with $x$ ranging from 0 to 1 and $y$ ranging from 0 to 1, we can set up the integral as follows:

$\iint_{\text{Region}} \frac{1}{1 + xy} \, dA$

This can be written as:

$\int_{0}^{1} \int_{0}^{1} \frac{1}{1 + xy} \, dx \, dy$

We can first integrate with respect to $x$ and then with respect to $y$. So, let's proceed with the integration:

1. Integrate with respect to $x$:

$\int_{0}^{1} \frac{1}{1 + xy} \, dx$

For this inner integral, we use the substitution $u = 1 + xy$, hence $du = y \, dx$. When $x = 0$, $u = 1$, and when $x = 1$, $u = 1 + y$. Therefore,

$dx = \frac{du}{y}$

Substituting into the integral, we get:

$\int_{1}^{1+y} \frac{1}{u} \cdot \frac{du}{y} = \frac{1}{y} \int_{1}^{1+y} \frac{1}{u} \, du = \frac{1}{y} \left[ \ln u \right]_{1}^{1+y}$

Evaluating the definite integral:

$\frac{1}{y} \left( \ln(1 + y) - \ln 1 \right) = \frac{\ln(1 + y)}{y}$

2. Integrate with respect to $y$:

Now we need to integrate $\frac{\ln(1 + y)}{y}$ with respect to $y$ from 0 to 1:

$\int_{0}^{1} \frac{\ln(1 + y)}{y} \, dy$

This integral is known as the Euler integral of the second kind and can be solved by recognizing it as the dilogarithm function, specifically $\text{Li}_2$:

$\int_{0}^{1} \frac{\ln(1 + y)}{y} \, dy = \text{Li}_2(2) - \text{Li}_2(1) = \frac{\pi^2}{6}$

Putting it all together, the value of the double integral is:

$\int_{0}^{1} \int_{0}^{1} \frac{1}{1 + xy} \, dx \, dy = \frac{\pi^2}{6}$

Summary

The double integral of $\frac{1}{1 + xy}$ over the region where $x$ and $y$ range from 0 to 1 is $\frac{\pi^2}{6}$.

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Related Questions

1. How do you evaluate the double integral $\iint_{\text{Region}} e^{xy} \, dA$ over the same region?
2. What is the meaning of the dilogarithm function $\text{Li}_2$?
3. How does the substitution method help in evaluating integrals?
4. Can you explain the steps to compute $\int_{0}^{1} \frac{\ln(1 + y)}{y} \, dy$ in more detail?
5. How would you evaluate the triple integral of $\frac{1}{1 + x y z}$ over $0 \leq x, y, z \leq 1$?
6. What is the significance of the Euler integral of the second kind in mathematics?
7. How do you determine the bounds for double integrals in different regions?
8. How does changing the order of integration affect the result of a double integral?

Tip

Understanding the relationship between integrals and known functions, like the dilogarithm, can simplify complex integral computations.