Question:
Find the integral:
\[
\int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx
\]
Find the integral:
\[
\int \left( \sin^{-1} \sqrt{x} + \cos^{-1} \sqrt{x} \right) \, dx
\]
Show Hint
When dealing with inverse trigonometric functions, use known identities to simplify the expression before performing the integration.
Updated On: Apr 28, 2025
- \( \frac{x}{2} + C \)
- \( x + C \)
- \( \frac{2x}{3} + C \)
- \( \frac{x^2}{2} + C \)
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The Correct Option is A
Solution and Explanation
We begin by recognizing that the sum of \( \sin^{-1} \sqrt{x} \) and \( \cos^{-1} \sqrt{x} \) is equal to \( \frac{\pi}{2} \), due to the identity \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \). Therefore, the integral becomes:
\[
\int \left( \frac{\pi}{2} \right) \, dx = \frac{\pi}{2} x + C
\]
Thus, the solution is \( \frac{x}{2} + C \), which is the correct answer.
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