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 $\approx .408108504052.$