To integrate the expression \((x - 4)(x + 4) \, dx\), let's first simplify the expression by expanding the product. Then, we can integrate the resulting polynomial.

### Step-by-Step Solution

1. **Expand the product:**

   \[
   (x - 4)(x + 4) = x^2 - 4^2 = x^2 - 16
   \]

   This is a difference of squares, which simplifies to \(x^2 - 16\).

2. **Set up the integral:**

   Now, we integrate the simplified expression:

   \[
   \int (x^2 - 16) \, dx
   \]

3. **Integrate each term separately:**

   - For \(\int x^2 \, dx\):

     \[
     \int x^2 \, dx = \frac{x^{2+1}}{2+1} = \frac{x^3}{3}
     \]

   - For \(\int (-16) \, dx\):

     \[
     \int (-16) \, dx = -16x
     \]

4. **Combine the integrals:**

   Now, we combine the integrated terms:

   \[
   \int (x^2 - 16) \, dx = \frac{x^3}{3} - 16x + C
   \]

   where \(C\) is the constant of integration.

### Final Answer:

\[
\int (x - 4)(x + 4) \, dx = \frac{x^3}{3} - 16x + C
\]

Would you like more details or have any questions?

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Here are 5 related questions you might be interested in:

1. How do you integrate a function that involves a sum or difference of polynomials?
2. What is the integration rule for a product of two linear terms?
3. How do you approach integrating polynomials of higher degrees?
4. What are some common integration techniques used for more complex expressions?
5. How does the Fundamental Theorem of Calculus relate to finding the integral of a function?

**Tip:** When expanding and simplifying expressions before integration, check if you can use algebraic identities (like the difference of squares) to make the integration process easier.

Learn how to integrate the expression (x - 4)(x + 4) with a detailed step-by-step solution. This problem covers integration of polynomials, specifically the difference of squares technique. Suitable for college-level mathematics students.

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