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To solve the integral

\[
\int (x^3 - 6x^2 + 3) \, dx
\]

we’ll integrate each term individually. Let's go through the solution:

1. **Integrate \(x^3\):**
   \[
   \int x^3 \, dx = \frac{x^{4}}{4}
   \]

2. **Integrate \(-6x^2\):**
   \[
   \int -6x^2 \, dx = -6 \cdot \frac{x^3}{3} = -2x^3
   \]

3. **Integrate \(3\):**
   \[
   \int 3 \, dx = 3x
   \]

Putting it all together, the integral becomes:

\[
\frac{x^4}{4} - 2x^3 + 3x + C
\]

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

So, the correct answer is:

\[
\frac{x^4}{4} - 2x^3 + 3x + C
\]

Would you like to see how to verify this by differentiation?

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### Additional Practice Questions
1. How would you integrate \( \int (2x^3 - 4x + 5) \, dx \)?
2. What is the antiderivative of \( x^5 \)?
3. How do you verify an integral solution by differentiation?
4. If you integrate \( -5x^4 + 2x \), what is the result?
5. What happens if there’s a coefficient before the integral, such as \( 2 \int (x^3 - 6x^2 + 3) \, dx \)?

### Tip:
When integrating, remember to add \(+C\) at the end to account for the constant of integration.

Evaluate the integral \(\int (x^3 - 6x^2 + 3) \, dx\) and check your answer by differentiating.

Learn how to evaluate the integral \(\int (x^3 - 6x^2 + 3) \, dx\) by finding the antiderivative of each term and verifying the result through differentiation. This solution covers key calculus concepts like integration and the Fundamental Theorem of Calculus. Suitable for advanced high school and college-level calculus students.

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