Improper Integral and Comparison Theorem

I think this is my third post regarding the Comparison theorem. I can rarely tell if my reasoning is correct when using this theorem. Please LMK if I'm wrong. I want determine if the following is convergent:

$\displaystyle \int_0^{\infty}\frac{Arctan(x)}{2+e^x}dx$

$\displaystyle x\geq 0$

$\displaystyle 0<Arctan(x)<\frac{\pi}{2}$

$\displaystyle \frac{Arctan(x)}{2+e^x}<\frac{\pi}{4+2e^x}<\frac{\ pi}{4}$

Since $\displaystyle \lim_{t->\infty}\int_0^t\frac{\pi}{4}dx$ diverges, the integral $\displaystyle \int_0^{\infty}\frac{Arctan(x)}{2+e^x}dx$ diverges by the Comparison Theorem.

(Thinking)