`int(1-x)sqrt(x) dx`

Answer

To solve the integral \( \int (1 - x) \sqrt{x} \, dx \), we can break it down into simpler parts. Here are the steps:

### Step 1: Expand the integrand
We start by expanding the integrand:
\[ \int (1 - x) \sqrt{x} \, dx = \int \sqrt{x} \, dx - \int x \sqrt{x} \, dx \]

### Step 2: Rewrite the integrals
Next, we rewrite \( \sqrt{x} \) and \( x \sqrt{x} \) in terms of powers of \( x \):
\[ \sqrt{x} = x^{1/2} \quad \text{and} \quad x \sqrt{x} = x^{1} \cdot x^{1/2} = x^{3/2} \]
Thus, we can express the integral as:
\[ \int \sqrt{x} \, dx - \int x^{3/2} \, dx = \int x^{1/2} \, dx - \int x^{3/2} \, dx \]

### Step 3: Integrate each term
Now we can integrate each term separately. The formula for integrating \( x^n \) is:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]

**For the first integral:**
\[ \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]

**For the second integral:**
\[ \int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2} \]

### Step 4: Combine the results
Now we combine the results of the integrals:
\[ \int (1 - x) \sqrt{x} \, dx = \frac{2}{3} x^{3/2} - \frac{2}{5} x^{5/2} + C \]

### Final Answer
Thus, the final answer is:
\[ \int (1 - x) \sqrt{x} \, dx = \frac{2}{3} x^{3/2} - \frac{2}{5} x^{5/2} + C \]