Assertion (A) : ∫9−x21dx=sin−13x. Reason (R) : ∫a2−x21dx=sin−1ax+C.
Get the complete, step-by-step math solution for: "Assertion (A) : ∫ {1}{√(9 - x²)} dx = sin^{-1} (x)/(3). Reason (R) : ∫ {1}{√(a² - x²)} dx = sin^{-1} (x)/(a) + C.". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the standard integral form
The given integral in the assertion, ∫9−x21dx, matches the standard integral form provided in the reason, ∫a2−x21dx. This standard form is a fundamental result in integral calculus.
Step 2: Compare and identify 'a'
By comparing the given integral ∫9−x21dx with the standard form ∫a2−x21dx, we can identify that a2=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−1ax+C. This gives us the result for the specific integral in the assertion.
Step 4: Evaluate the assertion and reason
The assertion states that ∫9−x21dx=sin−13x. Our calculation confirms this, so the assertion is true. The reason provides the general formula ∫a2−x21dx=sin−1ax+C, which is also true and directly used to derive the assertion. Therefore, the reason correctly explains the assertion.