Let's analyze and solve the given integrals.

We are given the following information: $\int_{2}^{6} f(x) \, dx = 10 \quad \text{and} \quad \int_{2}^{6} g(x) \, dx = -2$ We need to find the values of the integrals in parts (a) to (d).

(a) $\int_{2}^{6} [f(x) + g(x)] \, dx$

By the property of integrals: $\int_{2}^{6} [f(x) + g(x)] \, dx = \int_{2}^{6} f(x) \, dx + \int_{2}^{6} g(x) \, dx$ Substitute the given values: $= 10 + (-2) = 8$

(b) $\int_{2}^{6} [g(x) - f(x)] \, dx$

Again, using the linearity of integrals: $\int_{2}^{6} [g(x) - f(x)] \, dx = \int_{2}^{6} g(x) \, dx - \int_{2}^{6} f(x) \, dx$ Substitute the values: $= (-2) - 10 = -12$

(c) $\int_{2}^{6} 2g(x) \, dx$

Using the property of constants in integrals: $\int_{2}^{6} 2g(x) \, dx = 2 \int_{2}^{6} g(x) \, dx$ Substitute the value: $= 2(-2) = -4$

(d) $\int_{2}^{6} [2f(x) - 3g(x)] \, dx$

Again using the linearity of integrals and constants: $\int_{2}^{6} [2f(x) - 3g(x)] \, dx = 2 \int_{2}^{6} f(x) \, dx - 3 \int_{2}^{6} g(x) \, dx$ Substitute the values: $= 2(10) - 3(-2) = 20 + 6 = 26$

Final Answers:

• (a) 8
• (b) -12
• (c) -4
• (d) 26

Would you like further details or explanations on any part?

Here are 5 related questions:

1. How do you find the integral of a sum or difference of two functions?
2. What are the properties of definite integrals when constants are involved?
3. Can the result of a definite integral ever be negative, and why?
4. How do you interpret the result of an integral geometrically?
5. What happens if the limits of integration are swapped?

Tip: When dealing with integrals involving constants, factor them out to simplify the computation.