Question

Evaluate the integral: \[ \int_0^{\frac{\pi}{2}} \frac{\sin \left( \frac{\pi}{4} + x \right) + \sin \left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} \, dx \]

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

Hint:

Use trigonometric identities to simplify the sum of sines in the numerator and the expression in the denominator before evaluating the integral.

Answer 1

The Correct Option is B

We are given the integral: \[ \int_0^{\frac{\pi}{2}} \frac{\sin \left( \frac{\pi}{4} + x \right) + \sin \left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} \, dx \] Step 1: Use the sum of sines identity \( \sin A + \sin B = 2 \sin \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \) to simplify the numerator. Step 2: Simplify the denominator using the identity \( \cos x + \sin x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right) \). Step 3: The integral becomes a simpler trigonometric integral, and after solving, we find the result: \[ \frac{\pi}{2\sqrt{2}} \] % Final Answer The value of the integral is \( \frac{\pi}{2\sqrt{2}} \).