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Recognize the standard integral form:
\(\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}(\frac{x}{a}) + C\)
Apply the formula to the given integral:
\(\int_{0}^{3}\frac{dx}{\sqrt{9-x^{2}}} = \int_{0}^{3}\frac{dx}{\sqrt{3^2-x^{2}}} = \left[ \sin^{-1}(\frac{x}{3}) \right]_{0}^{3}\)
Substitute the limits of integration:
\(\left[ \sin^{-1}(\frac{x}{3}) \right]_{0}^{3} = \sin^{-1}(\frac{3}{3}) - \sin^{-1}(\frac{0}{3}) = \sin^{-1}(1) - \sin^{-1}(0)\)
Evaluate the inverse sine functions:
\(\sin^{-1}(1) = \frac{\pi}{2}\) and \(\sin^{-1}(0) = 0\)
Calculate the final result:
\(\frac{\pi}{2} - 0 = \frac{\pi}{2}\)
Correct Answer: \(\frac{\pi}{2}\)