Question

Evaluate: \[ \int_0^{\pi/2} \frac{x}{\sqrt{4 - x^2}} \, dx \]

• A. \( \frac{2}{3} \sin \left( \frac{x^2}{2} \right) \)
• B. \( \frac{3}{2} \sin \left( \frac{x^2}{3} \right) \)
• C. \( \frac{1}{2} \sin \left( \frac{x^2}{2} \right) \)
• D. None of these

Hint:

Use substitution to simplify integrals with square roots, and trigonometric identities to solve them.

Answer 1

The Correct Option is A

Step 1: Use trigonometric substitution.
Using the substitution \( x = 2 \sin \theta \), we evaluate the integral and find the solution to be \( \frac{2}{3} \sin \left( \frac{x^2}{2} \right) \).
Thus, the correct answer is (1).