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$\large \displaystyle \int_{{-\pi}/{2}}^{{\pi}/{2}} {\dfrac{ e^{\left| \sin x \right|} \cos x}{1 + e^{\tan x}}} \,dx ~ = ~ ?$

Notations: $e \approx 2.718$ is the Euler's number. For your answer take this approximate value as the value of $e$.

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