TITLE: convergence of $\int _1^{\infty} \sin\big(\mathrm{e}^x(x-2)\big)\,dx$
QUESTION [6 upvotes]: Question:
$$\int _1^{\infty} \sin\big(\mathrm{e}^x(x-2)\big)\,dx$$
does this converge?
Wolfram|Alpha doesn't have an answer, and I would really know. We tried to use Dirichlet and substituting with $t=e^x$. But couldn't continue

REPLY [2 votes]: It suffices to show the convergence of 
$$\int _3^{\infty} \sin\big(\mathrm{e}^x(x-2)\big)\,dx$$
This is of the form
$$\int _a^{\infty} \sin\big(f(x))\,dx$$
Note that we have 
$$\int _a^{N} \sin\big(f(x))\,dx = \int _a^{N} (f'(x)\sin\big(f(x))) \times {1 \over f'(x)}\,dx$$
Now integrate by parts in the right-hand expression, then let $N$ go to infinity...