I want to investigate the absolute convergence of integral. $$\int_{0}^{\infty} \; x^4 \; \sin(e^{2x}) \; dx$$ I made a replacement $$t = e^{2x},\; x = \frac{\ln{t}}{2} \\ \int_{1}^{\infty} \; \frac{\ln^4{t}}{32t} \; \sin{t} \;dt$$ I do not know how to continue. Please, help me
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2Even $\int_{1}^{\infty} ; \frac1{32t} ; \sin{t} ;dt$ is not absolutely convergent. – Anne Bauval May 28 '23 at 18:08
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Hint for (non-absolute) convergence: $\sin=-\cos'$ + Bertrand's integral. – Anne Bauval May 28 '23 at 20:21