Determine if the improper integral converges

Question

I'm having trouble proving that this improper integral converges, if it does.

$$\int_3^{\infty} \frac{dx}{x+e^x}$$

Answer 1

Observe that $0 < f(x) = \frac{1}{x+e^x} < g(x) := \frac{1}{e^x}$ for $x\in [3,\infty)$.

This implies that

$0\leq \int_3^\infty f(x)dx \leq \int_3^\infty g(x)dx$.

You can easily show that $\int_3^\infty g(x)dx$ is finite.