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!