Are these functions Lebesgue Integrable over these domains?

Question

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• $\frac{\sin x^2}{x}$ on $(1,\infty)$;
• $\frac1x\sin\frac{1}{x^2}$ on $(0,1)$.

Hi,

I've managed to prove that $\frac{\sin(x)}{x}$ is integrable on $(0,R]$ and is not Lebesgue integrable on $(1, \infty)$. Is there any way of using this to show whether the two functions above are or aren't Lebesgue integrable?

Thank you!

Answer 1

Try substituting $u=x^2$ in the first case and $u=1/x^2$ in the second case. I think you will get $\sin(u)/u$ in both cases.