Question

$\int(\sin^{-1}\sqrt{x}+\cos^{-1}\sqrt{x})dx=$ ________.

• A. $\frac{\pi}{2}+C$
• B. $\frac{\pi x}{4}+C$
• C. $\frac{\pi x}{3}+C$
• D. $\frac{\pi x}{2}+C$
• E. $\frac{-\pi x}{2}+C$

Hint:

Inverse trigonometric sums equal to $\pi/2$ are common simplifiers.

Answer 1

The Correct Option is D

Step 1: Concept
Use the inverse trigonometric identity $\sin^{-1}\theta + \cos^{-1}\theta = \frac{\pi}{2}$.

Step 2: Meaning
The integral simplifies to $\int \frac{\pi}{2} dx$.

Step 3: Analysis
Since $\frac{\pi}{2}$ is a constant, it can be taken out: $\frac{\pi}{2} \int 1 dx$.

Step 4: Conclusion
$\frac{\pi}{2}(x) + C = \frac{\pi x}{2} + C$. Final Answer: (D)