Evaluate: `int_0^1 1/(sqrt(1+x)+sqrt(x))dx`

Explanation

To evaluate the integral \( I = \int_0^1 \frac{1}{\sqrt{1+x} + \sqrt{x}} \, dx \), we will follow a systematic approach.

### Step 1: Rationalize the Denominator
We start by rationalizing the denominator. We multiply and divide by the conjugate of the denominator:

\[ I = \int_0^1 \frac{\sqrt{1+x} - \sqrt{x}}{(\sqrt{1+x} + \sqrt{x})(\sqrt{1+x} - \sqrt{x})} \, dx \]
...