The value of _-/2^/211+e^ x,dx is:

Question

The value of $\displaystyle\int_{-\pi/2}^{\pi/2}\frac{1}{1+e^{\sin x}}\,dx$ is:

Accepted Answer

I = $$\int\limits_{{{ - \pi } \over 2}}^{{\pi  \over 2}} {{1 \over {1 + {e^{\sin x}}}}dx} $$ ....(1)

Replacing x with $$\left( {{\pi  \over 2} - {\pi  \over 2} + x} \right)$$, we get

I = $$\int\limits_{ - {\pi  \over 2}}^{{\pi  \over 2}} {{{dx} \over {1 + {e^{ - \sin x}}}}} $$

= $$\int\limits_{ - {\pi  \over 2}}^{{\pi  \over 2}} {{{{e^{\sin x}}dx} \over {1 + {e^{\sin x}}}}} $$ .....(2)

Adding (1) and (2), we get

2I = $$\int\limits_{ - {\pi  \over 2}}^{{\pi  \over 2}} {1dx} $$ = $$\left( {{\pi  \over 2} - \left( { - {\pi  \over 2}} \right)} \right)$$ = $$\pi $$

$$ \Rightarrow $$ I = $${{\pi  \over 2}}$$

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