Evaluate `int x^(1//3)(2+x^(1//2))^(2)dx`.

Answer

To evaluate the integral \( I = \int x^{1/3} (2 + x^{1/2})^2 \, dx \), we will follow these steps:

### Step 1: Expand the integrand
First, we expand the expression \( (2 + x^{1/2})^2 \):
\[ (2 + x^{1/2})^2 = 2^2 + 2 \cdot 2 \cdot x^{1/2} + (x^{1/2})^2 = 4 + 4x^{1/2} + x \]
Thus, we can rewrite the integral as:
\[ I = \int x^{1/3} (4 + 4x^{1/2} + x) \, dx \]

### Step 2: Distribute \( x^{1/3} \)
Now, we distribute \( x^{1/3} \) across the expanded terms:
\[ I = \int (4x^{1/3} + 4x^{1/3} \cdot x^{1/2} + x^{1/3} \cdot x) \, dx \]
This simplifies to:
\[ I = \int (4x^{1/3} + 4x^{(1/3 + 1/2)} + x^{(1/3 + 1)}) \, dx \]
Calculating the exponents:
\[ 1/3 + 1/2 = 5/6 \quad \text{and} \quad 1/3 + 1 = 4/3 \]
So the integral becomes:
\[ I = \int (4x^{1/3} + 4x^{5/6} + x^{4/3}) \, dx \]

### Step 3: Integrate each term
Now we can integrate each term separately:
1. For \( 4x^{1/3} \):
\[ \int 4x^{1/3} \, dx = 4 \cdot \frac{x^{1/3 + 1}}{1/3 + 1} = 4 \cdot \frac{x^{4/3}}{4/3} = 3x^{4/3} \]

2. For \( 4x^{5/6} \):
\[ \int 4x^{5/6} \, dx = 4 \cdot \frac{x^{5/6 + 1}}{5/6 + 1} = 4 \cdot \frac{x^{11/6}}{11/6} = \frac{24}{11} x^{11/6} \]

3. For \( x^{4/3} \):
\[ \int x^{4/3} \, dx = \frac{x^{4/3 + 1}}{4/3 + 1} = \frac{x^{7/3}}{7/3} = \frac{3}{7} x^{7/3} \]

### Step 4: Combine the results
Now we combine all the integrated terms:
\[ I = 3x^{4/3} + \frac{24}{11} x^{11/6} + \frac{3}{7} x^{7/3} + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result for the integral is:
\[ I = 3x^{4/3} + \frac{24}{11} x^{11/6} + \frac{3}{7} x^{7/3} + C \]