Evaluate the following integrals.
` int ( sqrt(x) - (1)/(sqrt(x)))^(2) dx`

Answer

To evaluate the integral \( I = \int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 \, dx \), we will follow these steps:

### Step 1: Expand the integrand
First, we expand the expression inside the integral:
\[ \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 = \left( \sqrt{x} \right)^2 - 2 \cdot \sqrt{x} \cdot \frac{1}{\sqrt{x}} + \left( \frac{1}{\sqrt{x}} \right)^2 \]
This simplifies to:
\[ x - 2 + \frac{1}{x} \]

### Step 2: Rewrite the integral
Now, we can rewrite the integral:
\[ I = \int \left( x - 2 + \frac{1}{x} \right) \, dx \]

### Step 3: Integrate term by term
Now we will integrate each term separately:
1. The integral of \( x \) is:
\[ \int x \, dx = \frac{x^2}{2} \]
2. The integral of \( -2 \) is:
\[ \int -2 \, dx = -2x \]
3. The integral of \( \frac{1}{x} \) is:
\[ \int \frac{1}{x} \, dx = \log |x| \]

Putting it all together, we have:
\[ I = \frac{x^2}{2} - 2x + \log |x| + C \]
where \( C \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:
\[ I = \frac{x^2}{2} - 2x + \log |x| + C \]

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