The exercise asks to study the convergence of integral $ \int_{\mathbb{R}^{2}} \sin(x-y)e^{-x^2 - y^2} $.
It's easy to see, as a function $f: A \subset \mathbb{R}^{2} \to \mathbb{R}$ is integrable, if and only if, the set of discontinuity points of $f$ has zero measure and the funcion $\vert f \vert $ is integrable in $A$, that $f(x, y):= \sin(x-y)e^{-x^2 - y^2} $ is integrable in $\mathbb{R}^2$ because $\vert f \vert \leq e^{-x^2 - y^2}$ and $e^{- x^2 - y^2}$ is integrable in $\mathbb{R}^{2}$.
Now, how to compute the integral $ \int_{\mathbb{R}^{2}} \sin(x-y)e^{-x^2 - y^2} $ ?
I've tried to use common changes of variables as polar coordinates or other linear ones, but it doesn't make things easier...
If you want to compute the antiderivative, it is not "too difficult" if you consider that $$\int\sin(x-y)\,e^{-(x^2+ y^2)}\,dx\, dy=\Im\left(\int e^{i(x-y)}\,e^{-(x^2+ y^2)}\,dx\, dy \right)$$
Now, the exponent is $$-(x^2-ix)-(y^2+iy)=-\left(x-\frac{i}{2}\right)^2-\left(y+\frac{i}{2}\right)^2-\frac{1}{2}$$ and you face gaussian integrals. So $$\int e^{i(x-y)}\,e^{-(x^2+ y^2)}\,dx\, dy =\frac{\pi }{4 \sqrt{e}}\, \text{erf}\left(x-\frac{i}{2} \right)\, \text{erf}\left(y+\frac{i}{2} \right)$$ Now, what is really unpleasant is to take the imaginary part of it.
It is doable, using for sure the $\text{erfi}(.)$ function