Past exam of the integration course of the University of Oxford / 2021 / 2 / c

The function is not Lebesgue integrable. Fix $0<\eta <\pi /2$ and restrict to $x\ge 1$, $0<y<\eta$. There $\cos y\ge \cos \eta >0$. Tonelli's theorem for the nonnegative absolute value gives the lower boundThe last integral diverges. Indeed, on $[k\pi ,k\pi +\pi /6]$, $|\cos 2x|\ge 1/2$; each such interval contributes at least a positive constant times $1/(k+1)$. The harmonic series diverges. Hence the integral of $|g|$ is infinite, regardless of possible signed cancellations. One cannot infer Lebesgue integrability by formally applying Fubini's theorem to the oscillatory integrand.

The function is Lebesgue integrable. The inequalities $|\sin u|\le \min \{u,1\}$ for $u>0$ give bounds that handle the two ends of the $x$-axis differently. For $0<x<1$,For $x\ge 1$,Tonelli's theorem therefore provesThe factors of sine suppress the near-axis contributions that caused divergence in part i.