`intz^(-1//3) " dz "`

Answer

To solve the integral \( \int z^{-\frac{1}{3}} \, dz \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the integral**:
We need to evaluate the integral:
\[ \int z^{-\frac{1}{3}} \, dz \]

2. **Use the power rule for integration**:
The power rule states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
where \( C \) is the constant of integration, and \( n \neq -1 \).

3. **Determine \( n \)**:
In our case, \( n = -\frac{1}{3} \).

4. **Apply the power rule**:
We can now apply the power rule:
\[ \int z^{-\frac{1}{3}} \, dz = \frac{z^{-\frac{1}{3} + 1}}{-\frac{1}{3} + 1} + C \]

5. **Simplify the exponent**:
Calculate \( -\frac{1}{3} + 1 \):
\[ -\frac{1}{3} + 1 = -\frac{1}{3} + \frac{3}{3} = \frac{2}{3} \]
Thus, we have:
\[ \int z^{-\frac{1}{3}} \, dz = \frac{z^{\frac{2}{3}}}{\frac{2}{3}} + C \]

6. **Simplify the fraction**:
Dividing by \( \frac{2}{3} \) is the same as multiplying by its reciprocal:
\[ \frac{z^{\frac{2}{3}}}{\frac{2}{3}} = z^{\frac{2}{3}} \cdot \frac{3}{2} = \frac{3}{2} z^{\frac{2}{3}} \]

7. **Final result**:
Therefore, the integral evaluates to:
\[ \int z^{-\frac{1}{3}} \, dz = \frac{3}{2} z^{\frac{2}{3}} + C \]

### Final Answer:
\[ \int z^{-\frac{1}{3}} \, dz = \frac{3}{2} z^{\frac{2}{3}} + C \]

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