Let's examine each of the given statements one by one:

a) $\int_a^b ( f(x) + g(x)) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx$

This statement is correct. It reflects the linearity property of integrals, which states that the integral of a sum is equal to the sum of the integrals. Mathematically:

$\int_a^b \left( f(x) + g(x) \right) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx$

b) $\int_a^b (f(x) - g(x)) \, dx = \int_a^b (g(x) - f(x)) \, dx$

This statement is incorrect. The integral of a difference is not the same as switching the order of the terms inside the integrand. Instead, the correct relationship should be:

$\int_a^b (f(x) - g(x)) \, dx = -\int_a^b (g(x) - f(x)) \, dx$

The negative sign comes from switching the order of the functions inside the integral.

c) $\int_a^b (f(x) \cdot g(x)) \, dx = \int_a^b f(x) \, dx \cdot \int_a^b g(x) \, dx$

This statement is incorrect. The integral of the product of two functions is not generally equal to the product of their integrals. There is no such simple rule, and in fact, calculating $\int_a^b (f(x) \cdot g(x)) \, dx$ generally requires a more sophisticated approach, such as using integration by parts or specific techniques depending on the functions involved.

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Thus, the correct answer is:

a)

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Would you like more details on any of the properties of integrals? Here are five follow-up questions:

1. How can you prove the linearity of definite integrals?
2. What is the geometric interpretation of the definite integral?
3. When is the integral of the product of two functions equal to the product of their integrals?
4. What is the difference between definite and indefinite integrals?
5. How do the properties of definite integrals change when you switch the limits of integration?

Tip:

When dealing with multiple functions under an integral, it's useful to remember properties like linearity and to avoid assuming that the integral of a product is simply the product of integrals.