The image contains a mathematical problem where you need to calculate the definite integral:

$\int_{1}^{4} (-3x^2 + 8x + 4) \, dx$

Solution:

Step 1: Find the indefinite integral

The indefinite integral of the given function is: $\int (-3x^2 + 8x + 4) \, dx = -x^3 + 4x^2 + 4x + C$

Step 2: Evaluate the definite integral

Now, substitute the limits of integration into the indefinite integral:

$\int_{1}^{4} (-3x^2 + 8x + 4) \, dx = \Big[-x^3 + 4x^2 + 4x\Big]_{1}^{4}$

Step 3: Compute at $x = 4$

$\Big[-x^3 + 4x^2 + 4x\Big]_{x=4} = -(4)^3 + 4(4)^2 + 4(4)$ $= -64 + 64 + 16 = 16$

Step 4: Compute at $x = 1$

$\Big[-x^3 + 4x^2 + 4x\Big]_{x=1} = -(1)^3 + 4(1)^2 + 4(1)$ $= -1 + 4 + 4 = 7$

Step 5: Subtract the results

$\int_{1}^{4} (-3x^2 + 8x + 4) \, dx = 16 - 7 = 9$

Final Answer:

$\boxed{9}$

Do you want further clarification or details?

Related Questions:

1. How do you compute indefinite integrals of polynomial functions?
2. What is the importance of definite integrals in applications?
3. How would this problem change if the limits of integration were reversed?
4. What are some practical examples where definite integrals are used?
5. Can we compute the same integral using numerical approximation methods?

Tip:

When solving definite integrals, always simplify the indefinite integral first before substituting the limits to avoid errors.