The integral of \( 2x^3 \) with respect to \( x \) is calculated as follows:

\[
\int 2x^3 \, dx
\]

Apply the power rule for integration, which states:

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

Here, \( n = 3 \), so:

\[
\int 2x^3 \, dx = 2 \cdot \frac{x^{4}}{4} + C = \frac{2}{4}x^4 + C = \frac{1}{2}x^4 + C
\]

Thus, the result is:

\[
\frac{1}{2}x^4 + C
\]

Would you like more details or have any questions?

Here are five related questions:
1. What is the integral of \( 3x^2 \)?
2. How do you apply the power rule to a general polynomial?
3. What happens if you integrate \( x^n \) where \( n = -1 \)?
4. Can you explain how definite integrals differ from indefinite integrals?
5. How do constants affect the integration process?

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

Discover how to calculate the integral of 2x^3 using the power rule for integration. This problem demonstrates key calculus concepts such as indefinite integration, application of the power rule, and the importance of the constant of integration. Suitable for students in Grades 11-12.

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