To solve the problem, we need to evaluate:

$\int_4^8 \big[ 4f(x) + g(x) \big] \, dx$

Using the linearity of integrals:

$\int_4^8 \big[ 4f(x) + g(x) \big] \, dx = 4 \int_4^8 f(x) \, dx + \int_4^8 g(x) \, dx$

From the given information:

• $\int_4^8 f(x) \, dx = -6$
• $\int_4^8 g(x) \, dx = 10$

Substituting these values:

$4 \int_4^8 f(x) \, dx = 4(-6) = -24$ $\int_4^8 g(x) \, dx = 10$

Adding these together:

$\int_4^8 \big[ 4f(x) + g(x) \big] \, dx = -24 + 10 = -14$

Final Answer:

$\int_4^8 \big[ 4f(x) + g(x) \big] \, dx = -14$

Let me know if you want further clarification or details on the solution!

Here are 5 related questions for practice:

1. How would the solution change if $\int_4^8 f(x) dx = 0$?
2. What is the integral of $\int_4^8 \big[ 2f(x) - g(x) \big] dx$ using the same given values?
3. How does the linearity property of integrals simplify these types of problems?
4. Can you compute $\int_4^8 \big[ f(x) + g(x) \big] dx$ with the given data?
5. What if $f(x)$ and $g(x)$ were multiplied instead of added; how would the solution process differ?

Tip: Always verify properties of the function,