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
2026-05-16 01:45:03.1778895903
convergence of $\int _1^{\infty} \sin\big(\mathrm{e}^x(x-2)\big)\,dx$
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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...