To solve the given integral:

$\int_{11}^{-4} [4f(x) - 3g(x)] \, dx,$

we need to make use of the properties of integrals and the given values:

$\int_{-4}^{11} f(x) \, dx = 34$ $\int_{-4}^{11} g(x) \, dx = -5$

First, let's apply the property of integral limits. Changing the limits of integration will change the sign of the integral:

$\int_{a}^{b} h(x) \, dx = -\int_{b}^{a} h(x) \, dx$

Thus,

$\int_{11}^{-4} f(x) \, dx = -\int_{-4}^{11} f(x) \, dx$ $\int_{11}^{-4} f(x) \, dx = -34$

Similarly,

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

Now we can use the linearity of the integral to split it into parts:

$\int_{11}^{-4} [4f(x) - 3g(x)] \, dx = 4 \int_{11}^{-4} f(x) \, dx - 3 \int_{11}^{-4} g(x) \, dx$

Substituting the values we have calculated:

$4 \int_{11}^{-4} f(x) \, dx = 4(-34) = -136$ $-3 \int_{11}^{-4} g(x) \, dx = -3(5) = -15$

Adding these results together:

$\int_{11}^{-4} [4f(x) - 3g(x)] \, dx = -136 - 15 = -151$

Therefore, the value of the integral is:

$\int_{11}^{-4} [4f(x) - 3g(x)] \, dx = -151$

Would you like any further details or have any questions?

Here are 5 more questions you might find interesting:

1. How do you find the derivative of a product of two functions?
2. What are the rules for integration by parts?
3. How can you determine the convergence of an improper integral?
4. What is the Fundamental Theorem of Calculus?
5. How do you solve a differential equation using separation of variables?

Tip: When solving integrals, always consider whether you can simplify the problem using symmetry, linearity, or known values of integrals over the same interval.