To solve the given indefinite integral:

$\int (x - 4)^2 \, dx$

Step-by-step solution:

1. Expand the square: Expand $(x - 4)^2$ to simplify the integrand: $(x - 4)^2 = x^2 - 8x + 16$

   So, the integral becomes: $\int (x^2 - 8x + 16) \, dx$

2. Separate the terms: Break the integral into separate terms: $\int x^2 \, dx - 8 \int x \, dx + 16 \int 1 \, dx$

3. Integrate each term:

   • The integral of $x^2$ is $\frac{x^3}{3}$,
   • The integral of $x$ is $\frac{x^2}{2}$,
   • The integral of a constant $1$ is $x$.

   Substituting these: $\frac{x^3}{3} - 8 \cdot \frac{x^2}{2} + 16x$

4. Simplify the coefficients: Simplify $-8 \cdot \frac{x^2}{2}$ to $-4x^2$, resulting in: $\frac{x^3}{3} - 4x^2 + 16x + C$

Final Answer:

$\int (x - 4)^2 \, dx = \frac{x^3}{3} - 4x^2 + 16x + C$

Where $C$ is the constant of integration.

Would you like more details or have any further questions? Here are 5 related questions to expand understanding:

1. How does expanding $(x - 4)^2$ affect the integration process?
2. Why is a constant of integration ($C$) necessary in indefinite integrals?
3. How can substitution be used to solve similar integrals?
4. What happens if the integrand includes a higher-degree polynomial, e.g., $(x - 4)^3$?
5. How is this method different from numerical integration?

Tip: Always expand and simplify expressions before integrating, as it makes calculations clearer and avoids errors.