Assertion (A) : $\int \frac{1}{\sqrt{9 - x^2}} dx = \sin^{-1} \frac{x}{3}$. Reason (R) : $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1} \frac{x}{a} + C$.

Step 1: Identify the standard integral form

The given integral in the assertion, $\int \frac{1}{\sqrt{9 - x^2}} dx$, matches the standard integral form provided in the reason, $\int \frac{1}{\sqrt{a^2 - x^2}} dx$. This standard form is a fundamental result in integral calculus.

Step 2: Compare and identify 'a'

By comparing the given integral $\int \frac{1}{\sqrt{9 - x^2}} dx$ with the standard form $\int \frac{1}{\sqrt{a^2 - x^2}} dx$, we can identify that $a^2 = 9$. Taking the square root, we find that $a = 3$.

Step 3: Substitute 'a' into the standard formula

Now, substitute the value of $a=3$ into the standard integral formula $\sin^{-1} \frac{x}{a} + C$. This gives us the result for the specific integral in the assertion.

Step 4: Evaluate the assertion and reason

The assertion states that $\int \frac{1}{\sqrt{9 - x^2}} dx = \sin^{-1} \frac{x}{3}$. Our calculation confirms this, so the assertion is true. The reason provides the general formula $\int \frac{1}{\sqrt{a^2 - x^2}} dx = \sin^{-1} \frac{x}{a} + C$, which is also true and directly used to derive the assertion. Therefore, the reason correctly explains the assertion.

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