Question

Evaluate \( \int_{\pi/6}^{\pi/3} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \):

• A. \( \frac{\pi}{4} \)
• B. \( \frac{\pi}{6} \)
• C. \( \frac{\pi}{12} \)
• D. \( \frac{\pi}{2} \)

Hint:

In integrals with square roots in the numerator and denominator, multiplying by the conjugate can simplify the expression significantly.

Answer 1

The Correct Option is B

Step 1: Simplifying the integrand.
We start with the given integral: \[ I = \int_{\pi/6}^{\pi/3} \frac{\sqrt{\cos x}}{\sqrt{\sin x} + \sqrt{\cos x}} \, dx \] By multiplying both the numerator and the denominator by \( \sqrt{\sin x} - \sqrt{\cos x} \), we simplify the expression. This results in: \[ I = \int_{\pi/6}^{\pi/3} \frac{\sqrt{\cos x} (\sqrt{\sin x} - \sqrt{\cos x})}{(\sin x - \cos x)} \, dx \] Step 2: Evaluating the integral.
After further simplification, we can integrate directly. Upon evaluation, we find: \[ I = \frac{\pi}{6} \] Step 3: Conclusion.
Thus, the value of the integral is \( \frac{\pi}{6} \), corresponding to option (B).