## Step 1: Recognize the Integral Form The given integral is: \[ \int_^2 \frac{1}{\sqrt{9-x^2}} \, dx \] Notice that this matches the standard form: \[ \int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsinleft(\frac{x}{a}\right) + C \] Here, \( a = 3 \). **Explanation:** We use the arcsine rule for integrals involving the square root of a squared constant minus \( x^2 \). --- ## Step 2: Evaluate the Definite Integral Apply the formula: \[ \int_^2 \frac{1}{\sqrt{9-x^2}} dx = \left. \arcsin\left(\frac{x}{3}\right) \right|_^2 \] Calculate the values: - At \( x = 2 \): \( \arcsin\left(\frac{2}{3}\right) \) - At \( x = \): \( \arcsin() = \) Subtract: \[ \arcsin\left(\frac{2}{3}\right) - = \arcsin\left(\frac{2}{3}\right) \] **Explanation:** We plugged in the bounds and subtracted as required for definite integrals. --- ## **Final Answer** \[ \boxed{\arcsin\left(\frac{2}{3}\right)} \] Or, as a decimal (rounded): \[ \arcsin\left(\frac{2}{3}\right) \approx .7297 \]

## Step 1: Recognize the Integral Form The given integral is: \[ \int_0^2 \frac{1}{\sqrt{9-x^2}} \, dx \] This matches the standard form: \[ \int \frac{1}{\sqrt{a^2-x^2}} \, dx = \arcsin\left(\frac{x}{a}\right) + C \] where \( a = 3 \). **Explanation:** The integral represents the area under the curve of a semicircle with radius 3, which allows us to employ the arcsine function for evaluation. --- ## Step 2: Evaluate the Definite Integral Using the formula: \[ \int_0^2 \frac{1}{\sqrt{9-x^2}} \, dx = \left. \arcsin\left(\frac{x}{3}\right) \right|_0^2 \] Calculate the values: - At \( x = 2 \): \( \arcsin\left(\frac{2}{3}\right) \) - At \( x = 0 \): \( \arcsin\left(\frac{0}{3}\right) = 0 \) Now subtract: \[ \arcsin\left(\frac{2}{3}\right) - 0 = \arcsin\left(\frac{2}{3}\right) \] **Explanation:** We evaluated the antiderivative at the upper limit and the lower limit, then performed the subtraction as required for definite integrals. --- ## Final Answer \[ \boxed{\arcsin\left(\frac{2}{3}\right)} \] Or, as a decimal (approximately): \[ \arcsin\left(\frac{2}{3}\right) \approx 0.7297 \]