We are asked to find the indefinite integral of:

$\int \frac{x^2}{x - 9} \, dx$

To solve this, we can use long division to simplify the integrand.

Step 1: Long Division of $\frac{x^2}{x - 9}$

Perform the division of $x^2$ by $x - 9$:

$x^2 \div (x - 9) = x + 9 + \frac{81}{x - 9}$

So, we can rewrite the integral as:

$\int \left(x + 9 + \frac{81}{x - 9} \right) dx$

Step 2: Integrate Term by Term

Now we can integrate each term:

1. $\int x \, dx = \frac{x^2}{2}$
2. $\int 9 \, dx = 9x$
3. $\int \frac{81}{x - 9} \, dx = 81 \ln |x - 9|$

Step 3: Combine the Results

The solution to the integral is:

$\frac{x^2}{2} + 9x + 81 \ln |x - 9| + C$

Where $C$ is the constant of integration.

Would you like further details or have any questions about the steps?

Here are some related questions to expand on this concept:

1. How does long division of polynomials work in general?
2. Can this method be applied to rational functions with higher-degree polynomials?
3. How can you recognize when to use logarithms in integrals?
4. What other techniques exist for integrating rational functions?
5. What is the importance of the constant of integration in indefinite integrals?

Tip: Always check if the degree of the numerator is greater than or equal to the denominator before deciding to use long division in integrals.