Question

The value of \( \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \) is:

• A. $\pi/2$
• B. $\pi/4$
• C. $\pi$
• D. 0

Hint:

For integrals of the form $\frac{f(x)}{f(x)+f(a-x)}$ from $0$ to $a$, the answer is always $a/2$.

Answer 1

The Correct Option is B

Step 1: Concept
Use the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$.
Step 2: Analysis
Let $I = \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} dx$. Then $I = \int_{0}^{\pi/2} \frac{\sin(\pi/2 - x)}{\sin(\pi/2 - x) + \cos(\pi/2 - x)} dx = \int_{0}^{\pi/2} \frac{\cos x}{\cos x + \sin x} dx$. Adding the two equations: $2I = \int_{0}^{\pi/2} \frac{\sin x + \cos x}{\sin x + \cos x} dx = \int_{0}^{\pi/2} 1 dx$.
Step 3: Conclusion
$2I = [x]_{0}^{\pi/2} = \pi/2 \Rightarrow I = \pi/4$.
Final Answer: (B)