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The problem asks for the antiderivative of the function:

\[
\frac{7}{5} x^{8/5}
\]

To find the antiderivative, we need to apply the power rule for integration, which is given by:

\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad \text{(for } n \neq -1\text{)}
\]

### Steps:
1. The exponent in the function is \( n = \frac{8}{5} \).
2. Using the power rule, increase the exponent by 1:
   \[
   n+1 = \frac{8}{5} + 1 = \frac{8}{5} + \frac{5}{5} = \frac{13}{5}
   \]
3. Divide by the new exponent \( \frac{13}{5} \). This is equivalent to multiplying by \( \frac{5}{13} \):
   \[
   \int \frac{7}{5} x^{8/5} \, dx = \frac{7}{5} \cdot \frac{5}{13} x^{13/5} = \frac{7}{13} x^{13/5} + C
   \]

### Final Answer:
The correct option is \( \boxed{D} \), which is \( \frac{7}{13} x^{13/5} \).

Would you like a further breakdown of these steps or have any questions?

Here are some follow-up questions for practice:
1. What is the antiderivative of \( x^{5/3} \)?
2. How do you find the antiderivative of \( \frac{3}{2} x^{7/4} \)?
3. What happens when you integrate a negative power of \( x \), like \( x^{-3/2} \)?
4. How does the constant factor rule affect integration?
5. Why can't we use the power rule when the exponent is \( -1 \)?

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

Find an antiderivative of the function 7/5 * x^(8/5).

Learn how to find the antiderivative of 7/5 * x^(8/5) using the power rule of integration. This step-by-step guide is perfect for students learning pre-calculus or calculus concepts, focusing on key integration techniques. Suitable for Grades 11-12.

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