What is \int_{0}^{\frac{\pi}{2}}\frac{a+\sin x}{2a+\sin x+\cos x}\,dx equal to ?
- A. \frac{\pi}{4} ✓
- B. \frac{\pi}{2}
- C. 1
- D. 0
Correct Answer: A. \frac{\pi}{4}
Explanation
Let I = \int_{0}^{\frac{\pi}{2}}\frac{a+\sin x}{2a+\sin x+\cos x}dx. Using the property \int_0^a f(x)dx = \int_0^a f(a-x)dx, I = \int_{0}^{\frac{\pi}{2}}\frac{a+\cos x}{2a+\cos x+\sin x}dx. Adding both equations gives 2I = \int_{0}^{\frac{\pi}{2}}\frac{2a+\sin x+\cos x}{2a+\sin x+\cos x}dx = \int_{0}^{\frac{\pi}{2}} 1 dx = \frac{\pi}{2}. Thus, I = \frac{\pi}{4}.
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