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

Explanation

To solve the integral \( \int \frac{dx}{x - \sqrt{x}} \), we will follow these steps:

### Step 1: Factor out \( \sqrt{x} \)
We start by factoring \( \sqrt{x} \) from the denominator:

\[ x - \sqrt{x} = \sqrt{x}(\sqrt{x} - 1) \]

Thus, we can rewrite the integral as:

\[ \int \frac{dx}{\sqrt{x}(\sqrt{x} - 1)} \]

### Step 2: Substitute \( u = \sqrt{x} - 1 \)
Next, we make the substitution \( u = \sqrt{x} - 1 \). Therefore, we have:

\[ \sqrt{x} = u + 1 \quad \text{and} \quad x = (u + 1)^2 \]

Now, we differentiate \( x \) with respect to \( u \):

\[ dx = 2(u + 1) du \]

### Step 3: Rewrite the integral in terms of \( u \)
Substituting \( \sqrt{x} \) and \( dx \) into the integral, we get:

\[ \int \frac{2(u + 1) du}{(u + 1)(u)} = \int \frac{2 du}{u} \]

### Step 4: Integrate
Now we can integrate:

\[ \int \frac{2 du}{u} = 2 \ln |u| + C \]

### Step 5: Substitute back for \( u \)
We substitute back \( u = \sqrt{x} - 1 \):

\[ 2 \ln |\sqrt{x} - 1| + C \]

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

\[ \int \frac{dx}{x - \sqrt{x}} = 2 \ln |\sqrt{x} - 1| + C \]

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