[Math] Using Comparison test to determine if $\int_0^{\infty} \frac{\arctan x} {2+e^{x}} \ dx$ converges

Question

Only using the Comparison test, I am trying to see if the following integral converges: $$\int_0^{\infty} \frac{\arctan x} {2+e^{x}} \ dx$$

I first noted that $\arctan x \lt (2+e^{x}) \ \forall x \in \mathbb{R}$ which allows me to say that

$$\int_0^{\infty} \frac{\arctan x} {2+e^{x}} \ dx \lt \infty$$

I'm not sure where to progress from here though.

Mathematica reports the integral converging to $\approx .408108504052.$

Answer 1

HINT

$\vert\arctan(x)\vert \in \left[ 0, \pi/2\right]$ and $2+e^x > e^x$. Can you now finish it off?