Question

The value of \( \int_0^{\frac{\pi}{2}} \frac{\sin\left( \frac{\pi}{4} + x \right) + \sin\left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} \, dx \) is:

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

Hint:

When dealing with integrals involving trigonometric sums, use sum-to-product identities to simplify the numerator.

Answer 1

The Correct Option is B

Step 1: Let \( I = \int_0^{\frac{\pi}{2}} \frac{\sin\left( \frac{\pi}{4} + x \right) + \sin\left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} dx \). We will simplify the integral using trigonometric identities. \[I = \int_0^{\frac{\pi}{2}} \frac{\sin\left( \frac{\pi}{4} + x \right) + \sin\left( \frac{3\pi}{4} + x \right)}{\cos x + \sin x} dx\]Applying the sum-to-product identity to the numerator yields:\[I = \int_0^{\frac{\pi}{2}} \frac{2\sin\left(\frac{\pi}{2} + x \right)}{\cos x + \sin x} dx\]Step 2: Further simplification using the identity \( \sin\left(\frac{\pi}{2} + x \right) = \cos x \) results in:\[I = \int_0^{\frac{\pi}{2}} \frac{2\cos x}{\cos x + \sin x} dx\]Substitution and trigonometric simplifications will be used to find the solution. The value of the integral is \( \frac{\pi}{2\sqrt{2}} \).