$\int_{-4}^{4} \int_{-2}^{0} \frac{30000 e^{y}}{1+\frac{|x|}{5}} d y d x$

Step 1 ：First, we need to understand the problem. We are asked to evaluate a double integral. The outer integral is with respect to \(x\) and ranges from \(-4\) to \(4\), and the inner integral is with respect to \(y\) and ranges from \(-2\) to \(0\). The function to be integrated is \(\frac{30000 e^{y}}{1+\frac{|x|}{5}}\).

Step 2 ：We notice that the function is symmetric with respect to \(x\), i.e., the function does not change if we replace \(x\) with \(-x\). Therefore, we can simplify the calculation by considering the integral from \(0\) to \(4\) and then doubling the result.

Step 3 ：We start by calculating the inner integral: \(\int_{-2}^{0} \frac{30000 e^{y}}{1+\frac{|x|}{5}} d y\).

Step 4 ：The antiderivative of \(\frac{30000 e^{y}}{1+\frac{|x|}{5}}\) with respect to \(y\) is \(\frac{30000 e^{y}}{1+\frac{|x|}{5}}\).

Step 5 ：We evaluate this antiderivative at the limits of integration: \(\frac{30000 e^{0}}{1+\frac{|x|}{5}} - \frac{30000 e^{-2}}{1+\frac{|x|}{5}} = \frac{30000}{1+\frac{|x|}{5}} - \frac{30000 e^{-2}}{1+\frac{|x|}{5}}\).

Step 6 ：We simplify this expression to get: \(30000 - 30000 e^{-2}\).

Step 7 ：Now we calculate the outer integral: \(\int_{0}^{4} (30000 - 30000 e^{-2}) d x\).

Step 8 ：The antiderivative of \(30000 - 30000 e^{-2}\) with respect to \(x\) is \(30000x - 30000 e^{-2}x\).

Step 9 ：We evaluate this antiderivative at the limits of integration: \(30000*4 - 30000 e^{-2}*4 - (30000*0 - 30000 e^{-2}*0) = 120000 - 120000 e^{-2}\).

Step 10 ：We double this result because we only considered the integral from \(0\) to \(4\), not from \(-4\) to \(4\). So the final result is \(2*(120000 - 120000 e^{-2}) = 240000 - 240000 e^{-2}\).

Step 11 ：We check that this result meets the requirements of the problem. The integral was calculated correctly, and the result is in its simplest form.

Step 12 ：So, the final answer is \(\boxed{240000 - 240000 e^{-2}}\).