The given integral is:

$\int (x^2 - 9x) \, dx$

To solve this, we can split the integral into two parts:

$\int x^2 \, dx - \int 9x \, dx$

Step 1: Evaluate each integral separately.

1. For $\int x^2 \, dx$:

$\int x^2 \, dx = \frac{x^3}{3}$

2. For $\int 9x \, dx$:

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

Step 2: Combine the results.

Now, subtract the two integrals:

$\frac{x^3}{3} - \frac{9x^2}{2} + C$

Where $C$ is the constant of integration.

Final answer:

$\frac{x^3}{3} - \frac{9x^2}{2} + C$

Do you want more details or have any questions?

Here are 5 related questions to expand on this topic:

1. How do you handle integrals of polynomials in general?
2. What happens if there is a constant multiplier in the integrand?
3. How do you solve integrals involving higher powers of $x$?
4. What are the general rules for finding antiderivatives?
5. How would you compute the definite integral of this function from 0 to 2?

Tip: Always remember to include the constant of integration $C$ when solving indefinite integrals.