Evaluate the definite integral $\int_{0}^{1} \frac{dx}{\sqrt{1+x}-\sqrt{x}}$.

(D) Let $I = \int_{0}^{1} \frac{dx}{\sqrt{1+x}-\sqrt{x}}$.
Rationalizing the denominator:
$I = \int_{0}^{1} \frac{1}{(\sqrt{1+x}-\sqrt{x})} \times \frac{(\sqrt{1+x}+\sqrt{x})}{(\sqrt{1+x}+\sqrt{x})} dx$
$I = \int_{0}^{1} \frac{\sqrt{1+x}+\sqrt{x}}{(1+x)-x} dx$
$I = \int_{0}^{1} (\sqrt{1+x} + \sqrt{x}) dx$
$I = \left[ \frac{2}{3}(1+x)^{3/2} \right]_{0}^{1} + \left[ \frac{2}{3}x^{3/2} \right]_{0}^{1}$
$I = \frac{2}{3} \left[ (1+1)^{3/2} - (1+0)^{3/2} \right] + \frac{2}{3} [1^{3/2} - 0^{3/2}]$
$I = \frac{2}{3} [2^{3/2} - 1] + \frac{2}{3} [1]$
$I = \frac{2}{3} (2\sqrt{2} - 1) + \frac{2}{3}$
$I = \frac{4\sqrt{2}}{3} - \frac{2}{3} + \frac{2}{3} = \frac{4\sqrt{2}}{3}$.