Find the following integrals :
`int (1-x) sqrtx dx`

Answer

To solve the integral \( \int (1-x) \sqrt{x} \, dx \), we will first simplify the expression and then apply integration techniques.

### Step-by-Step Solution:

1. **Expand the Integrand**:
\[ \int (1-x) \sqrt{x} \, dx = \int \left( \sqrt{x} - x \sqrt{x} \right) \, dx \]
Here, we can rewrite \( x \sqrt{x} \) as \( x^{3/2} \).

2. **Rewrite the Integral**:
\[ \int (1-x) \sqrt{x} \, dx = \int \sqrt{x} \, dx - \int x^{3/2} \, dx \]

3. **Integrate Each Term**:
- For the first term \( \int \sqrt{x} \, dx \):
\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2} \]
- For the second term \( \int x^{3/2} \, dx \):
\[ \int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2} \]

4. **Combine the Results**:
Now we can combine the results of the two integrals:
\[ \int (1-x) \sqrt{x} \, dx = \frac{2}{3} x^{3/2} - \frac{2}{5} x^{5/2} + C \]
where \( C \) is the constant of integration.

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