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The problem asks you to calculate the integral of the function \( ax - x^2 \) with respect to \( x \). This can be expressed mathematically as:

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

### Solution:
To solve this integral, we'll break it down into two separate integrals:

\[
\int ax \, dx - \int x^2 \, dx
\]

1. **First Integral: \( \int ax \, dx \)**

   The integral of \( ax \) with respect to \( x \) is calculated using the power rule:

   \[
   \int ax \, dx = \frac{a}{2}x^2 + C_1
   \]

   Here, \( C_1 \) is a constant of integration.

2. **Second Integral: \( \int x^2 \, dx \)**

   Again, using the power rule for integration:

   \[
   \int x^2 \, dx = \frac{1}{3}x^3 + C_2
   \]

   Here, \( C_2 \) is another constant of integration.

### Putting it all together:

\[
\int (ax - x^2) \, dx = \frac{a}{2}x^2 - \frac{1}{3}x^3 + C
\]

Where \( C = C_1 - C_2 \) is the combined constant of integration.

Thus, the final answer is:

\[
\boxed{\frac{a}{2}x^2 - \frac{1}{3}x^3 + C}
\]

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

### Related Questions:
1. How do you integrate functions involving higher powers of \(x\)?
2. What is the geometric interpretation of an integral?
3. How do you handle integrals with additional parameters (like \(a\))?
4. What are the differences between definite and indefinite integrals?
5. How can integration be applied in physics and engineering problems?

### Tip:
When integrating polynomial functions, always apply the power rule for each term individually, and remember to add the constant of integration at the end!

Learn how to solve the integral ∫ (ax - x^2) dx by applying the power rule of integration. This step-by-step solution is suitable for high school students in Grades 11-12.

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