Question

\( \int_{0}^{\pi/2} \frac{2\sin x}{2\sin x + 2\cos x} dx \)

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

Hint:

Use substitution \( x \to \frac{\pi}{2}-x \) in definite integrals.

Answer 1

The Correct Option is C

Concept: Use symmetry property \( f(x)+f(\frac{\pi}{2}-x) \)

Step 1: Simplify integrand: \[ \frac{2\sin x}{2(\sin x+\cos x)} = \frac{\sin x}{\sin x+\cos x} \]

Step 2: Let: \[ I = \int_0^{\pi/2} \frac{\sin x}{\sin x+\cos x} dx \]

Step 3: Replace \( x \to \frac{\pi}{2}-x \): \[ I = \int_0^{\pi/2} \frac{\cos x}{\sin x+\cos x} dx \]

Step 4: Add both: \[ 2I = \int_0^{\pi/2} \frac{\sin x+\cos x}{\sin x+\cos x} dx = \int_0^{\pi/2} 1 dx \]

Step 5: Evaluate: \[ 2I = \frac{\pi}{2} \Rightarrow I = \frac{\pi}{4} \]