Question

\textbf \( \int_{0}^{\pi/2} \frac{\sqrt{\sin x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \) is equal to:}

• A. \( 0 \)
• B. \( -\pi \)
• C. \( \frac{3\pi}{2} \)
• D. \( \frac{\pi}{2} \)
• E. \( \frac{\pi}{4} \)

Hint:

For integrals of the form \( \int_{0}^{\pi/2} \frac{f(\sin x)}{f(\sin x) + f(\cos x)} \, dx \), the result is always half of the upper limit, which is \( \pi/4 \). This applies to powers, roots, or even logarithmic functions of sin and cos.

Answer 1

Answer: (E)