The problem asks to evaluate the integral:

$\int_{-3}^{5} g(x) \, dx$

given the following information:

$\int_{-3}^{0} g(x) \, dx = 5, \quad \int_{0}^{3} g(x) \, dx = 2, \quad \text{and} \quad \int_{3}^{5} g(x) \, dx = 4.$

To solve this, we break the integral $\int_{-3}^{5} g(x) \, dx$ into three parts:

$\int_{-3}^{5} g(x) \, dx = \int_{-3}^{0} g(x) \, dx + \int_{0}^{3} g(x) \, dx + \int_{3}^{5} g(x) \, dx.$

From the problem, we are provided with these values:

• $\int_{-3}^{0} g(x) \, dx = 5$,
• $\int_{0}^{3} g(x) \, dx = 2$,
• $\int_{3}^{5} g(x) \, dx = 4$.

Now, we sum these values:

$\int_{-3}^{5} g(x) \, dx = 5 + 2 + 4 = 11.$

Thus, the value of $\int_{-3}^{5} g(x) \, dx$ is 11.

Would you like further details or clarification? Here are 5 related questions to explore further:

1. What does it mean to break an integral into smaller intervals?
2. How do definite integrals handle discontinuous functions across different intervals?
3. Can this technique be applied to more complex functions with unknown integrals?
4. What is the geometric interpretation of definite integrals?
5. How do properties like linearity of integrals help solve more complex integral problems?

Tip: Always check if an integral can be broken into smaller parts when provided with partial integral information, as it simplifies the computation.