The value of the integral $\int_{0}^{4} \frac{1}{1 + \sqrt{x}} \, dx$ is $4 - 2 \ln(3)$.

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Here are 5 related questions to explore:

1. How do you compute definite integrals with square root functions?
2. What is the significance of logarithms in integration results?
3. Can integrals like this be solved using substitution techniques?
4. What are the rules for integrating functions involving fractional powers?
5. How do improper integrals differ from definite integrals?

Tip: In cases where an integral involves a square root, a substitution like $u = \sqrt{x}$ often simplifies the process.