The problem involves evaluating the definite integral:

$\int_{-3}^{3} \left(4x^3 + 9\right) \, dx$

Step 1: Apply the properties of definite integrals

We notice the integral is split into two parts: $\int_{-3}^{3} \left(4x^3 + 9\right) \, dx = \int_{-3}^{3} 4x^3 \, dx + \int_{-3}^{3} 9 \, dx$

Symmetry Property:

• $4x^3$ is an odd function ($f(-x) = -f(x)$), and the integral of an odd function over a symmetric interval $[-a, a]$ is 0: $\int_{-3}^{3} 4x^3 \, dx = 0$

Constant Term:

• $9$ is a constant, so: $\int_{-3}^{3} 9 \, dx = 9 \int_{-3}^{3} 1 \, dx = 9 \cdot \left[ x \right]_{-3}^{3}$

Step 2: Evaluate each integral

1. Odd function term: $\int_{-3}^{3} 4x^3 \, dx = 0$

2. Constant term: $\int_{-3}^{3} 9 \, dx = 9 \cdot \left[ x \right]_{-3}^{3} = 9 \cdot \left(3 - (-3)\right) = 9 \cdot 6 = 54$

Step 3: Combine results

$\int_{-3}^{3} \left(4x^3 + 9\right) \, dx = 0 + 54 = 54$

Final Answer:

$\boxed{54}$

Would you like further clarification or detailed steps?

Here are 5 related questions to expand your understanding:

1. How does the symmetry property simplify definite integrals?
2. Can you identify other examples of odd and even functions?
3. What happens if we change the limits of integration to asymmetric intervals?
4. How does the integral of a constant function behave for different limits?
5. How can definite integrals be used in real-world applications?

Tip: When solving definite integrals, always check for symmetry; it can save a lot of time in evaluation.