The value of the integral, computed to four decimal places, is approximately:

\[
\boxed{0.7360}
\]

Would you like any more details or have any questions?

Here are 8 related questions to expand on this topic:
1. How do you determine the Jacobian determinant for other transformations?
2. What are some common change of variables used in double integrals?
3. Can you explain the concept of a Jacobian matrix in multivariable calculus?
4. How do you handle integrals over irregular regions without transformation?
5. What are some applications of double integrals in physics and engineering?
6. How do you evaluate double integrals in polar coordinates?
7. What are the properties of logarithmic functions used in integration?
8. How do you confirm the accuracy of your integral calculations?

**Tip:** When performing calculations that involve logarithms or other transcendental functions, it's useful to keep a list of common values and their approximations handy to speed up the process.

Explore how to calculate the value of an integral approximation using methods in calculus. This solution provides an approximate value of 0.7360 computed to four decimal places. Learn more about integral approximation and related concepts in advanced calculus.

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