# 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\$\$