Determine whether $f=(sin x^2)/x$ is (Lebesgue) integrable on $(1,infty)$.

I am given that $sin x/x$ is not lebesgue integrable on $(1,infty)$ so I guess I would need to find a connection between the two functions $sin x/x$ and $sin x^2/x$.

My guess is that in fact $f$ is not integrable, so I tried to find a lower bound function of $|f|$ which is not integrable, and tried rather obvious $(sin x^2)/x^2$, but in the end my attempt failed at some stage when I substituted $t=x^2$ and changed integral into something involving $sin t/t$.

or maybe I could use change of variable $x=1/t$ to convert the function into $tsin (1/t^2)$, since $sin(1/t^2)$ is a nice(!) function. But it doesnt seem to work as well.

and now I’m really stuck… I cannot seem to think of any other idea.

Any helps appreciated!

Answer

It seems you already tried this. Perhaps you didn’t realize what you got? Let $u=t^2$. Then $du=2tdt$ and $du/u=2dt/t$, so that $$int_1^{infty}sin(t^2)t^{-1}dt=frac 1 2int_1^infty frac{sin u}udu$$

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