Definite Integration
General
Grade 12
Question:
<p>$\int_{0}^{\pi/2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ equals -</p>
2
\pi
\frac{\pi}{4}
\frac{\pi}{2}
Step-by-Step Solution
Key Concept: General
<div>$I = \int_{0}^{\pi/2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx = \int_{0}^{\pi/2} \frac{2^{\sin(\pi/2 - x)}}{2^{\sin(\pi/2 - x)} + 2^{\cos(\pi/2 - x)}} dx = \int_{0}^{\pi/2} \frac{2^{\cos x}}{2^{\cos x} + 2^{\sin x}} dx$<br>$2I = \int_{0}^{\pi/2} dx = \frac{\pi}{2} \Rightarrow I = \frac{\pi}{4}$</div>
Correct Answer: C