To evaluate the integral

$\int \left( x^{\frac{2}{3}} - x^{\frac{5}{3}} \right) dx,$

we integrate each term separately:

1. The integral of $x^{\frac{2}{3}}$: $\int x^{\frac{2}{3}} dx = \frac{x^{\frac{2}{3} + 1}}{\frac{2}{3} + 1} = \frac{x^{\frac{5}{3}}}{\frac{5}{3}} = \frac{3}{5}x^{\frac{5}{3}}.$

2. The integral of $x^{\frac{5}{3}}$: $\int x^{\frac{5}{3}} dx = \frac{x^{\frac{5}{3} + 1}}{\frac{5}{3} + 1} = \frac{x^{\frac{8}{3}}}{\frac{8}{3}} = \frac{3}{8}x^{\frac{8}{3}}.$

Now combine the results: $\int \left( x^{\frac{2}{3}} - x^{\frac{5}{3}} \right) dx = \frac{3}{5}x^{\frac{5}{3}} - \frac{3}{8}x^{\frac{8}{3}} + C.$

The correct option is not explicitly visible in the provided choices but follows this result. If you'd like clarification or further verification, feel free to ask!

Would you like a detailed step-by-step explanation for the steps? Here are five related questions for further exploration:

1. How does one handle negative exponents when integrating?
2. How can you verify an indefinite integral by differentiation?
3. What are the rules for integrating fractional exponents?
4. How does adding a constant $C$ affect the solution?
5. Can the same process be used to integrate $x^{n} - x^{m}$ for general $n$ and $m$?

Tip: Always simplify the exponent calculations carefully when dealing with fractional exponents!