$\int_a^\infty x^r\sin(x)\,dx$ diverges for all $r>0$

How can I prove that for all $a,r>0$ the integral $\int_a^\infty x^r\sin(x)\,dx$ diverges?

I first tried a limit comparison test with the divergent $\int_a^\infty x^r\,dx= \infty$.

This leads to the divergent limit:

$$\lim_{x\to\infty}|\sin x |$$Which means that the test was indecisive.

Another idea was to get a recursive formula for the integral, using integration by parts. And then to use induction to prove divergence starting from $r=0$. But this can't be easily extended to non-integer $r$.

Thanks in advance for any help.