Convergence of an integral, without evaluating it

Question

I'm trying to prove the convergence of this integral, but I can't figure out the solution.

$\int_{0}^{\infty}{\sin(e^x)}dx$

Answer 1

If we put $$e^x=t $$

it has the same nature than

$$\int_1^{+\infty}\frac {\sin (t)}{t}dt $$ which converges by Dirichlet's criteria or by parts integration.