Evaluate the integrals.
`int 2 x sqrt(x)` dx

Answer

To evaluate the integral \( \int 2x \sqrt{x} \, dx \), we can follow these steps:

### Step 1: Rewrite the Integrand
We start by rewriting \( \sqrt{x} \) in terms of exponents:
\[ \sqrt{x} = x^{1/2} \]
Thus, we can rewrite the integrand:
\[ 2x \sqrt{x} = 2x \cdot x^{1/2} = 2x^{1 + 1/2} = 2x^{3/2} \]

### Step 2: Set Up the Integral
Now, we can set up the integral:
\[ \int 2x \sqrt{x} \, dx = \int 2x^{3/2} \, dx \]

### Step 3: Factor Out the Constant
We can factor out the constant \(2\) from the integral:
\[ = 2 \int x^{3/2} \, dx \]

### Step 4: Apply the Power Rule for Integration
Using the power rule for integration, which states that:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
we apply it here with \( n = \frac{3}{2} \):
\[ \int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} + C = \frac{x^{5/2}}{5/2} + C \]
This simplifies to:
\[ = \frac{2}{5} x^{5/2} + C \]

### Step 5: Substitute Back into the Integral
Now, substituting this back into our integral:
\[ 2 \int x^{3/2} \, dx = 2 \left( \frac{2}{5} x^{5/2} + C \right) = \frac{4}{5} x^{5/2} + 2C \]
Since \(2C\) is still a constant, we can denote it as \(C\):
\[ = \frac{4}{5} x^{5/2} + C \]

### Final Answer
Thus, the final answer for the integral is:
\[ \int 2x \sqrt{x} \, dx = \frac{4}{5} x^{5/2} + C \]

---