How can I prove that the following improper integral diverge?
$$\int_e^\infty y \sin \left( \frac{2}{y} \right) \ dx$$
I think I'm supposed to separate it into two parts, but I don't know how to do it. Any input?
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3$x\sin(1/x)\to1$ as $x\to\infty$ – Kenta S Feb 05 '21 at 21:31
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Put $$t=\frac 1x$$
Your integral has the same nature than $$\int_0^1\frac 1t\sin(t)\frac{dt}{t^2}$$
$$=\int_0^1\frac{\sin(t)dt}{t^3}$$
but, near $ 0^+ $,
$$\frac{\sin(t)}{t^3}\sim \frac{1}{t^2}$$
and $$\int_0^1\frac{dt}{t^2} \text{ diverges}$$
hamam_Abdallah
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It's not clear for me how my integral has the same nature as the one you first describe, i.e.1/t*sin(t) dt/t^2. Can you develope this? – John_Average Feb 05 '21 at 21:36
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@John_Average When $ t\to 0^+$ $\frac{\sin(t)}{t^3}\to \frac{1}{t^2}$. – hamam_Abdallah Feb 05 '21 at 22:27
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