Evaluate: `int(dx)/(sqrt(x)+sqrt(x-2))`

Explanation

To evaluate the integral

\[ I = \int \frac{dx}{\sqrt{x} + \sqrt{x-2}}, \]

we can start by rationalizing the denominator. This involves multiplying the numerator and denominator by the conjugate of the denominator, which is \(\sqrt{x} - \sqrt{x-2}\).

### Step 1: Rationalize the Denominator

Multiply and divide by \(\sqrt{x} - \sqrt{x-2}\):

\[ I = \int \frac{dx (\sqrt{x} - \sqrt{x-2})}{(\sqrt{x} + \sqrt{x-2})(\sqrt{x} - \sqrt{x-2})}. \]

### Step 2: Simplify the Denominator

The denominator simplifies as follows:

\[ (\sqrt{x} + \sqrt{x-2})(\sqrt{x} - \sqrt{x-2}) = x - (x - 2) = 2. \]

Thus, we can rewrite the integral:

\[ I = \int \frac{\sqrt{x} - \sqrt{x-2}}{2} \, dx = \frac{1}{2} \int (\sqrt{x} - \sqrt{x-2}) \, dx. \]

### Step 3: Split the Integral

Now we can split the integral into two parts:

\[ I = \frac{1}{2} \left( \int \sqrt{x} \, dx - \int \sqrt{x-2} \, dx \right). \]

### Step 4: Evaluate Each Integral

1. For \(\int \sqrt{x} \, dx\):

Using the formula \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\):

\[ \int \sqrt{x} \, dx = \int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}. \]

2. For \(\int \sqrt{x-2} \, dx\):

Using the same formula:

\[ \int \sqrt{x-2} \, dx = \int (x-2)^{1/2} \, dx = \frac{(x-2)^{3/2}}{3/2} = \frac{2}{3} (x-2)^{3/2}. \]

### Step 5: Combine Results

Now substituting back into our expression for \(I\):

\[ I = \frac{1}{2} \left( \frac{2}{3} x^{3/2} - \frac{2}{3} (x-2)^{3/2} \right). \]

This simplifies to:

\[ I = \frac{1}{3} x^{3/2} - \frac{1}{3} (x-2)^{3/2} + C. \]

### Final Answer

Thus, the final result is:

\[ I = \frac{1}{3} x^{3/2} - \frac{1}{3} (x-2)^{3/2} + C. \]

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