Question

The value of $\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{2^{\sin x}}{2^{\sin x}+2^{\cos x}}dx$ is

• A. $\frac{\pi }{4}$
• B. $\pi$
• C. $0$
• D. none of these

Answer 1

The correct option is A $\frac{\pi }{4}$

Let $I=$$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{2^{\sin x}}{2^{\sin x}+2^{\cos x}}dx$......(1).

or, $I=$$\underset{0}{\overset{\frac{\pi }{2}}{\int }}\frac{2^{\cos x}}{2^{\cos x}+2^{\sin x}}dx$ [ Using property of definite integral]......(2).

Now adding (1) ad (2) we get,

$2I=$$\underset{0}{\overset{\frac{\pi }{2}}{\int }}dx=\frac{\pi }{2}$

or, $I=\frac{\pi }{4}$.