= Solution
{c}
\b[The <function> is not <Lebesgue integrable>.] Fix $0<\eta<\pi/2$ and restrict to $x\geq1$, $0<y<\eta$. There $\cos y\geq\cos\eta>0$. <Tonelli's theorem> for the nonnegative absolute value gives the lower bound
$$
\begin{aligned}
\iint|g(x,y)|\,dx\,dy
&\geq\cos\eta\int_1^\infty|\cos2x|\int_0^\eta e^{-xy}\,dy\,dx\\
&=\cos\eta\int_1^\infty|\cos2x|\frac{1-e^{-\eta x}}x\,dx\\
&\geq\cos\eta(1-e^{-\eta})\int_1^\infty\frac{|\cos2x|}x\,dx.
\end{aligned}
$$
The last integral diverges. Indeed, on $[k\pi,k\pi+\pi/6]$, $|\cos2x|\geq1/2$; 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.
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