Step-by-Step Solution

1. Recognize the standard integral form:

   \(\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a}) + C\)

2. Apply the formula to the given integral:

   \(\int_{0}^{3}\frac{dx}{\sqrt{9-x^{2}}} = \int_{0}^{3}\frac{dx}{\sqrt{3^2-x^{2}}} = \left[ \sin^{-1}(\frac{x}{3}) \right]_{0}^{3}\)

3. Substitute the limits of integration:

   \(\left[ \sin^{-1}(\frac{x}{3}) \right]_{0}^{3} = \sin^{-1}(\frac{3}{3}) - \sin^{-1}(\frac{0}{3}) = \sin^{-1}(1) - \sin^{-1}(0)\)

4. Evaluate the inverse sine functions:

   \(\sin^{-1}(1) = \frac{\pi}{2}\) and \(\sin^{-1}(0) = 0\)

5. Calculate the final result:

   \(\frac{\pi}{2} - 0 = \frac{\pi}{2}\)