Question

Find the value of \( \displaystyle \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx \).

• A. \(\frac{\pi}{2}\)
• B. \(\frac{\pi}{8}\)
• C. \(\frac{\pi}{4}\)
• D. \(1\)

Hint:

For integrals of the form \(\int_0^{\pi/2} \frac{f(\sin x,\cos x)}{g(\sin x,\cos x)}dx\), try the substitution \(x \to \frac{\pi}{2}-x\) and add both expressions.

Answer 1

The Correct Option is C

Concept: For definite integrals, \[ \int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx \] This symmetry property helps simplify many trigonometric integrals.

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

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. \[ 2I=\left[x\right]_0^{\pi/2}=\frac{\pi}{2} \] \[ I=\frac{\pi}{4} \] \[ \boxed{\frac{\pi}{4}} \]