Simplify the fraction inside the square root.

$$\sqrt{\frac{1}{20}} = \frac{\sqrt{1}}{\sqrt{20}} = \frac{1}{\sqrt{4 \times 5}} = \frac{1}{2\sqrt{5}}$$

To rationalize the denominator, multiply the numerator and denominator by $\sqrt{5}$:

$$\frac{1}{2\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{2 \times 5} = \frac{\sqrt{5}}{10}$$

So the integral becomes:

$$\int_{0}^{1} \frac{\sqrt{5}}{10}\,dt$$

$$\left( \frac{\sqrt{5}}{10} \times 1 \right) - \left( \frac{\sqrt{5}}{10} \times 0 \right) = \frac{\sqrt{5}}{10} - 0 = \frac{\sqrt{5}}{10}$$

The final answer is $\frac{\sqrt{5}}{10}$.