The problem is the evaluation of the definite integral:

$\int_{-\ln 3}^{\ln 3} \frac{e^x}{e^x + 4} \, dx$

Solution:

1. Simplify the integral: Let $I = \int_{-\ln 3}^{\ln 3} \frac{e^x}{e^x + 4} \, dx$.

   Notice that this integral has limits symmetric about $0$ and the function can potentially exhibit symmetry. To check, substitute $x = -t$. Then $dx = -dt$, and the integral becomes: $I = \int_{-\ln 3}^{\ln 3} \frac{e^x}{e^x + 4} \, dx = \int_{\ln 3}^{-\ln 3} \frac{e^{-t}}{e^{-t} + 4} (-dt).$

2. Simplify the new expression: When switching the limits, the integral changes sign: $I = \int_{-\ln 3}^{\ln 3} \frac{e^{-t}}{e^{-t} + 4} \, dt.$

   Next, simplify $\frac{e^{-t}}{e^{-t} + 4}$: $e^{-t} = \frac{1}{e^t}, \quad \text{so} \quad \frac{e^{-t}}{e^{-t} + 4} = \frac{\frac{1}{e^t}}{\frac{1}{e^t} + 4} = \frac{1}{1 + 4e^t}.$

   Substituting this into the integral, we now have: $I = \int_{-\ln 3}^{\ln 3} \frac{1}{1 + 4e^t} \, dt.$

3. Combine original and transformed integrals: Notice the symmetry in the original integral: $\frac{e^x}{e^x + 4} + \frac{1}{1 + 4e^x} = 1.$

   Hence, the integral simplifies as follows: $I = \int_{-\ln 3}^{\ln 3} 1 \, dx - \int_{-\ln 3}^{\ln 3} \frac{1}{1 + 4e^x} \, dx.$

   Evaluate the first term: $\int_{-\ln 3}^{\ln 3} 1 \, dx = (\ln 3 - (-\ln 3)) = 2 \ln 3.$

   For the second term, note that due to symmetry, it equals half of the total integral. Hence: $I = 2 \ln 3 - I.$

4. Solve for $I$: Combine terms: $2I = 2 \ln 3 \implies I = \ln 3.$

Final Answer:

$\boxed{\ln 3}$

Would you like a detailed explanation of any step?
Here are 5 related questions for further exploration:

1. How do you test for symmetry in definite integrals?
2. What are the benefits of substitution techniques in integrals?
3. Can this integral be solved using a different substitution?
4. How would you solve this integral if the limits were not symmetric?
5. What role does the property $f(x) + f(-x) = c$ play in integral simplification?

Tip: Always look for symmetry in definite integrals—it can significantly simplify the problem!