Mathematical Expression:
The given expression is ∫_{0 to ∞} ∫_{0 to ∞} e^{-(y/x + 1) - x} dy dx.

Explanation:
To evaluate this double integral, we can first integrate with respect to y and then with respect to x.

Integration with respect to y:
Let's focus on the inner integral first, which involves integrating with respect to y:

∫_{0 to ∞} e^{-(y/x + 1) - x} dy

To evaluate this integral, we can use the substitution method. Let's substitute u = -(y/x + 1) - x, which implies du = -dy/x:

∫_{0 to ∞} e^u (-x du)

The limits of integration also change:

When y = 0, u = -(0/x + 1) - x = -1 - x.
When y = ∞, u = -(∞/x + 1) - x = -∞.

Now, the integral becomes:

∫_{-1-x to -∞} e^u (-x du)

To evaluate this integral, we can use the antiderivative of e^u, which is simply e^u:

-e^u ∣_{-1-x to -∞}
= -e^(-∞) - (-e^(-1-x))
= 0 - (-e^(-1-x))
= e^(-1-x).

Integration with respect to x:
Now that we have the result of the inner integral, let's integrate it with respect to x:

∫_{0 to ∞} e^(-1-x) dx

To evaluate this integral, we can use the antiderivative of e^(-1-x), which is -e^(-1-x):

-e^(-1-x) ∣_{0 to ∞}
= -e^(-1-∞) - (-e^(-1-0))
= 0 - (-e^(-1-0))
= e^(-1).

Final Result:
Therefore, the value of the given double integral is e^(-1).

Summary:
- The given expression is a double integral involving the exponential function.
- We first integrate with respect to y, using the substitution method.
- The resulting integral is then integrated with respect to x.
- Finally, the value of the double integral is e^(-1).