Question

Evaluate the integral \( \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \).

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{6} \)
• D. \( \frac{\pi}{3} \)

Hint:

For definite integrals involving \( \sin x \) and \( \cos x \), try replacing \( x \) with \( a - x \) to exploit symmetry and simplify the expression.

Answer 1

The Correct Option is B

Concept: Use the property of definite integrals: \[ \int_{0}^{a} f(x)\,dx = \int_{0}^{a} f(a-x)\,dx \] This is especially useful when symmetry simplifies the integrand.

Step 1: Let \( I = \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \). Using the property: \[ I = \int_{0}^{\pi/2} \frac{\cos x}{\sin x + \cos x} \, dx \]

Step 2: Add the two expressions. \[ 2I = \int_{0}^{\pi/2} \left( \frac{\sin x}{\sin x + \cos x} + \frac{\cos x}{\sin x + \cos x} \right) dx \] \[ 2I = \int_{0}^{\pi/2} 1 \, dx \]

Step 3: Evaluate the integral. \[ 2I = \left[ x \right]_{0}^{\pi/2} = \frac{\pi}{2} \] \[ I = \frac{\pi}{4} \]