Showing that $\int_{0}^{\pi/2}\frac{\tan x}{1+e^x}\,dx$ diverges.

Question

I want to show that the improper integral $\displaystyle\int_{0}^{\pi/2}\frac{\tan x}{1+e^x}\,dx$ diverges. I know that from $0$ to $\pi/2$ the following inequalities hold:

\begin{align*} \frac{\tan x}{1+e^x}&<\frac{\tan x}{2e^x}\\ \frac{\tan x}{1+e^x}&<\frac{x}{2e^x} \end{align*}

But I can't use them to show the integral diverges. Any help is appreciated.

Answer 1

$$\int_0^{π/2}\frac{\tan x}{1+e^x}\rm dx\ge\frac1{1+e^{π/2}}\int_0^{π/2}\tan x\rm dx$$ and the right hand side integral diverges.