Past exam of the options I course of the University of Oxford / 2008 / D / D2 / b

The function is measurable and $0\le {\cos }^2(1/x)\le 1$. The interval has finite measure, so its integral is at most one. Hence it is integrable, despite its oscillations near zero.

As $x\to \infty$, ${\cos }^2(1/x)\to 1$. It is therefore at least $1/2$ on some infinite tail. The integral of that lower bound is infinite, so it is not integrable on this interval.