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The integral you provided is:

\[
dy = 6 \sqrt[3]{x^2} \left( \frac{x}{2} - \frac{4}{3} \sqrt[3]{x^2} - \frac{1}{3} \right) dx
\]

We will solve this step-by-step.

### Step 1: Simplify the expression inside the parentheses
The integral can be rewritten as:

\[
dy = 6 \sqrt[3]{x^2} \left( \frac{x}{2} - \frac{4}{3} \sqrt[3]{x^2} - \frac{1}{3} \right) dx
\]

To simplify further, distribute the \(6 \sqrt[3]{x^2}\) term across the terms inside the parentheses:

\[
dy = 6 \left[ \sqrt[3]{x^2} \cdot \frac{x}{2} - \sqrt[3]{x^2} \cdot \frac{4}{3} \sqrt[3]{x^2} - \sqrt[3]{x^2} \cdot \frac{1}{3} \right] dx
\]

### Step 2: Simplify each term
- For the first term, \(\sqrt[3]{x^2} \cdot \frac{x}{2}\):
  \[
  \sqrt[3]{x^2} \cdot x = x^{\frac{2}{3}} \cdot x = x^{\frac{5}{3}}
  \]
  So the first term becomes:
  \[
  \frac{6}{2} x^{\frac{5}{3}} = 3 x^{\frac{5}{3}}
  \]

- For the second term, \(\sqrt[3]{x^2} \cdot \frac{4}{3} \sqrt[3]{x^2}\):
  \[
  \sqrt[3]{x^2} \cdot \sqrt[3]{x^2} = x^{\frac{2}{3}} \cdot x^{\frac{2}{3}} = x^{\frac{4}{3}}
  \]
  So the second term becomes:
  \[
  6 \cdot \frac{4}{3} \cdot x^{\frac{4}{3}} = 8 x^{\frac{4}{3}}
  \]

- For the third term, \(\sqrt[3]{x^2} \cdot \frac{1}{3}\):
  \[
  6 \cdot \frac{1}{3} \cdot \sqrt[3]{x^2} = 2 x^{\frac{2}{3}}
  \]

### Step 3: Combine the terms
Now the expression for \(dy\) becomes:

\[
dy = \left( 3 x^{\frac{5}{3}} - 8 x^{\frac{4}{3}} - 2 x^{\frac{2}{3}} \right) dx
\]

### Step 4: Integrate term by term
We can now integrate each term individually:

\[
\int dy = \int \left( 3 x^{\frac{5}{3}} - 8 x^{\frac{4}{3}} - 2 x^{\frac{2}{3}} \right) dx
\]

- For the first term: \(\int 3 x^{\frac{5}{3}} dx\):
  \[
  \frac{3}{\frac{5}{3} + 1} x^{\frac{5}{3} + 1} = \frac{3}{\frac{8}{3}} x^{\frac{8}{3}} = \frac{9}{8} x^{\frac{8}{3}}
  \]

- For the second term: \(\int -8 x^{\frac{4}{3}} dx\):
  \[
  \frac{-8}{\frac{4}{3} + 1} x^{\frac{4}{3} + 1} = \frac{-8}{\frac{7}{3}} x^{\frac{7}{3}} = -\frac{24}{7} x^{\frac{7}{3}}
  \]

- For the third term: \(\int -2 x^{\frac{2}{3}} dx\):
  \[
  \frac{-2}{\frac{2}{3} + 1} x^{\frac{2}{3} + 1} = \frac{-2}{\frac{5}{3}} x^{\frac{5}{3}} = -\frac{6}{5} x^{\frac{5}{3}}
  \]

### Step 5: Combine the results
Now combine the integrated terms:

\[
y = \frac{9}{8} x^{\frac{8}{3}} - \frac{24}{7} x^{\frac{7}{3}} - \frac{6}{5} x^{\frac{5}{3}} + C
\]

where \(C\) is the constant of integration.

This is the solution to the integral.

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Would you like any further details or clarifications on this solution?

Here are 5 related questions to explore:
1. How do we handle cube roots in calculus and simplify their powers?
2. What are the general rules for integrating terms with fractional exponents?
3. How can we verify the correctness of an integral solution?
4. How can constants affect the outcome of an integration problem?
5. What are common strategies for simplifying complex integrals?

**Tip:** When working with fractional exponents, converting between radicals and exponents can often simplify the process of integration.

Explore the step-by-step solution to solving an integral involving cube roots and fractional exponents. This problem demonstrates key calculus concepts, including simplification and integration rules for fractional powers.

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