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

Answer

To solve the integral \( \int \frac{1-x}{\sqrt{x}} \, dx \), we can break it down into simpler parts. Here’s the step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{1-x}{\sqrt{x}} \, dx \]
We can separate the terms in the numerator:
\[ I = \int \left( \frac{1}{\sqrt{x}} - \frac{x}{\sqrt{x}} \right) \, dx \]
This simplifies to:
\[ I = \int \left( \frac{1}{\sqrt{x}} - \sqrt{x} \right) \, dx \]

### Step 2: Integrate Each Term
Now we can integrate each term separately:
1. The integral of \( \frac{1}{\sqrt{x}} \):
\[ \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} \]
2. The integral of \( \sqrt{x} \):
\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{2}{3} x^{3/2} \]

### Step 3: Combine the Results
Putting it all together, we have:
\[ I = 2\sqrt{x} - \frac{2}{3} x^{3/2} + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the value of the integral is:
\[ \int \frac{1-x}{\sqrt{x}} \, dx = 2\sqrt{x} - \frac{2}{3} x^{3/2} + C \]