Question

$\int\limits^{\pi/2}_{0} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ equals

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

Answer 1

Answer: (C)

$\pi / 4$

Answer 2

$I = \int\limits^{\pi/2}_{0} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$
$I = \int\limits^{\pi/2}_{0} \frac{2^{\sin\left(\pi/2-x\right)}}{2^{\sin\left(\pi/2-x\right)} + 2^{\cos\left(\pi/2-x\right)}}dx$
$= \int \frac{2^{\cos x}}{2^{\cos x} + 2^{\sin x}} dx$
$\Rightarrow 21 = \int\limits^{\pi/2}_{0} dx = \frac{\pi}{2}$
$\Rightarrow 1 = \frac{\pi}{4}$