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How do I find $I =\int_0^{2\pi} ((e^{|\sin x|} \cos x)/(1+e^{\tan x})) \,dx$? The answer is $I =0$. I tried putting $2\pi-x$ instead of $x$ and got the same result except with $-\tan x$ in denominator. I'm not sure what to try next, would someone please help me with this?

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Decompose the integral as $$I = \int_0^\pi \frac{e^{\sin x}\cos x}{1+e^{\tan x}}dx + \int_{\pi}^{2\pi} \frac{e^{-\sin x}\cos x}{1+e^{\tan x}}dx.$$ Now use a change of variable $x\to x-\pi$ in the second integral and do the necessary changes in the integral to obtain $$I = \int_0^\pi \frac{e^{\sin x}\cos x}{1+e^{\tan x}}dx - \int_{0}^{\pi} \frac{e^{\sin x}\cos x}{1+e^{\tan x}}dx=0,$$ where in the last step we have used the elementary identities $\sin(x+\pi)=-\sin x,\ \cos(x+\pi)=-\cos x$ and,$\ \tan(x+\pi)=\tan x,\ \forall 0\le x\le \pi$.

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