Showing function is lebesgue integrable

Question

I have a function $f(x) = \frac{\sin(\frac{1}{x})}{1+(\log(x))^2}$ and I am trying to find whether this is Lebesgue integrable on $[1,\infty)$.

I'm really not sure where to start on this one. It seems like it perhaps should be lebesgue integrable because both $\sin(1/x)$ and $1+\log(x)^2$ are going to zero, but I know this is no guarantee.

Any general tips on showing whether a certain function is lebesgue integrable or not is also mush appreciated.

Thanks

Answer 1

Hint: For $1 \leqslant x$, we have

$$0 < \sin \frac{1}{x} < \frac{1}{x}.$$

In fact, $\sin \frac{1}{x} \sim \frac{1}{x}$ as $x\to \infty$, so the question is whether

$$\frac{1}{x(1+(\log x)^2)}$$

is integrable.