The value of int_{0}^{pi /2} frac{2^{sin x}}{ | vedclass

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(A) Let $I = \int_{0}^{\pi /2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ ....$(i)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x)

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$\frac{\pi}{4}$

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(A) Let $I = \int_{0}^{\pi /2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ ....$(i)$Using the property $\int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx$,we get:$I = \int_{0}^{\pi /2} \frac{2^{\sin(\pi/2 - x)}}{2^{\sin(\pi/2 - x)} + 2^{\cos(\pi/2 - x)}} dx$Since $\sin(\pi/2 - x) = \cos x$ and $\cos(\pi/2 - x) = \sin x$,we have:$I = \int_{0}^{\pi /2} \frac{2^{\cos x}}{2^{\cos x} + 2^{\sin x}} dx$ ....$(ii)$Adding equations $(i)$ and $(ii)$:$2I = \int_{0}^{\pi /2} \left( \frac{2^{\sin x} + 2^{\cos x}}{2^{\sin x} + 2^{\cos x}} \right) dx$$2I = \int_{0}^{\pi /2} 1 dx$$2I = [x]_{0}^{\pi /2} = \frac{\pi}{2}$$I = \frac{\pi}{4}$.

The value of $\int_{0}^{\pi /2} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} dx$ is

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