Find the following integrals :
`int (sqrtx-1/sqrtx)^2 dx`

Answer

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

### Step 1: Expand the integrand
First, we expand the expression inside the integral:

\[ \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 = \left( \sqrt{x} \right)^2 - 2 \left( \sqrt{x} \right) \left( \frac{1}{\sqrt{x}} \right) + \left( \frac{1}{\sqrt{x}} \right)^2 \]

This simplifies to:

\[ x - 2 + \frac{1}{x} \]

### Step 2: Rewrite the integral
Now we rewrite the integral with the expanded expression:

\[ \int \left( x - 2 + \frac{1}{x} \right) \, dx \]

### Step 3: Integrate term by term
Next, we integrate each term separately:

1. The integral of \( x \) is \( \frac{x^2}{2} \).
2. The integral of \( -2 \) is \( -2x \).
3. The integral of \( \frac{1}{x} \) is \( \ln |x| \).

Putting it all together, we have:

\[ \int \left( x - 2 + \frac{1}{x} \right) \, dx = \frac{x^2}{2} - 2x + \ln |x| + C \]

where \( C \) is the constant of integration.

### Final Answer
Thus, the final result of the integral is:

\[ \int \left( \sqrt{x} - \frac{1}{\sqrt{x}} \right)^2 \, dx = \frac{x^2}{2} - 2x + \ln |x| + C \]

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