https://file.aiquickdraw.com/mathbot/file/1725469249897b0gpvzey.jpg

The integral provided in the image is:

\[
\int (3x - x^4) \, dx
\]

To solve this, we'll integrate each term separately.

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

2. **Integrate \(-x^4\):**
   \[
   \int -x^4 \, dx = -\frac{x^5}{5}
   \]

Now, we combine these results:

\[
\int (3x - x^4) \, dx = \frac{3x^2}{2} - \frac{x^5}{5} + C
\]

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

Would you like more details or have any questions? Here are some related questions:

1. How do you differentiate the result to check the correctness of the integration?
2. What are the basic rules of integration used in this problem?
3. How does the power rule apply to the integration of polynomials?
4. What is the significance of the constant of integration \(C\)?
5. How would you approach integrating a function with more complex terms, such as trigonometric or exponential functions?

**Tip:** Always double-check your integration by differentiating your answer; the derivative should match the original integrand.

Explore the step-by-step process to integrate the polynomial function 3x - x^4. This detailed solution covers the use of the Power Rule for Integration, making it ideal for students in Grades 11-12.

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