Mathematics

The value of $$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\dfrac{2^{\sin x}}{2^{\sin x}+2^{\cos x}}dx$$ is


ANSWER

$$\dfrac{\pi}{4}$$


SOLUTION
Let $$I=$$$$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\dfrac{2^{\sin x}}{2^{\sin x}+2^{\cos x}}dx$$......(1).
or, $$I=$$$$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}\dfrac{2^{\cos x}}{2^{\cos x}+2^{\sin x}}dx$$ [ Using property of definite integral]......(2).
Now adding (1) ad (2) we get,
$$2I=$$$$\displaystyle\int\limits_{0}^{\dfrac{\pi}{2}}dx=\dfrac{\pi}{2}$$
or, $$I=\dfrac{\pi}{4}$$.
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Single Correct Medium Published on 17th 09, 2020
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