To find the integral of \( x^3 \), we will use the power rule of integration. The power rule states that:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\]

where \( n \neq -1 \) and \( C \) is the constant of integration.

For \( x^3 \):

\[
\int x^3 \, dx = \frac{x^{3+1}}{3+1} + C = \frac{x^4}{4} + C
\]

So, the integral of \( x^3 \) is:

\[
\boxed{\frac{x^4}{4} + C}
\]

Would you like more details or have any questions about this process? Here are some related questions you might find interesting:

1. What is the integral of \( x^2 \)?
2. How do you find the integral of \( x^n \) for any integer \( n \)?
3. What happens if \( n = -1 \) in the power rule?
4. How do you integrate polynomials with multiple terms?
5. Can you explain the concept of the constant of integration?
6. What is the integral of \( x^3 + 2x \)?
7. How do you check if an integral calculation is correct?
8. What are some applications of integrating \( x^3 \) in real-world problems?

**Tip:** Always remember to add the constant of integration \( C \) when finding indefinite integrals.

Explore how to find the integral of x^3 using the power rule of integration. This solution provides a detailed step-by-step process, including the application of the power rule and the constant of integration. Suitable for advanced high school students and calculus enthusiasts.

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