# $I =\int_0^{2\pi} ((e^{|\sin x|} \cos x)/(1+e^{\tan x})) \,dx$

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?

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,$$\ \tan(x+\pi)=\tan x,\ \forall 0\le x\le \pi$$.