Question

Find \(\int_{0}^{\frac{\pi}{2}} \sqrt{\cos x \sin 2x} \, dx\)

Hint:

For integrals involving square roots and trigonometric functions, look for possible trigonometric identities or substitutions to simplify the expression.

Answer 1

Step 1: Simplify the integrand.
Recall that \(\sin 2x = 2 \sin x \cos x\), so the integrand becomes: \[ \sqrt{\cos x \cdot 2 \sin x \cos x} = \sqrt{2 \cos^2 x \sin x} \] This simplifies to: \[ \sqrt{2} \cos x \sqrt{\sin x} \]
Step 2: Solve the integral.
Now, we need to compute: \[ \int_{0}^{\frac{\pi}{2}} \sqrt{2} \cos x \sqrt{\sin x} \, dx \] This can be solved using standard integration techniques or by substituting \(\sin x = t^2\) and solving the integral. After performing the integration, the result is: \[ \boxed{1} \]