Evaluate :
`int(sqrt(ax)-(1)/(sqrt(ax)))^(2)dx`

Answer

To evaluate the integral

\[ \int \left( \sqrt{ax} - \frac{1}{\sqrt{ax}} \right)^2 dx, \]

we will first expand the integrand and then integrate term by term.

### Step 1: Expand the integrand
Using the identity \((A - B)^2 = A^2 - 2AB + B^2\), we can expand the expression:

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

This simplifies to:

\[ ax - 2 + \frac{1}{ax}. \]

### Step 2: Write the integral
Now we can rewrite the integral:

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

### Step 3: Integrate term by term
Now we will integrate each term separately:

1. \(\int ax \, dx = \frac{a}{2} x^2\)
2. \(\int -2 \, dx = -2x\)
3. \(\int \frac{1}{ax} \, dx = \frac{1}{a} \ln |x|\)

Putting it all together, we have:

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

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

### Final Answer
Thus, the final answer is:

\[ \frac{a}{2} x^2 - 2x + \frac{1}{a} \ln |x| + C. \]

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