While solving
$$ \iint\limits_{(-\infty, +\infty)} \dfrac{\cos(x^2 + y^2)}{e^{x^2+ y^2}} \, \, dx dy$$
I reached the following stage after passing to polar coordinates:
$$ \iint\limits_{(-\infty, +\infty)} \dfrac{\cos(r^2)}{e^{r^2}} \, \, rdr \, d\theta = \dfrac12\iint\limits_{(-\infty, +\infty)} \dfrac{\cos(u)}{e^{u}} \, \, du \,d\theta $$
Now, since there is no $\theta$ variable in the integrand, if we choose to integrate first with respect to $\theta$ then our partial integral is automatically divergent. Thus my question is,
Is the divergence of at least one of the inner integrals in a multiple integral enough to conclude that the whole multiple integral diverges?
$$\iint_{\mathbb{R}^2}\frac{\cos(x^2+y^2)}{\exp(x^2+y^2)}\,dx\,dy = 2\pi\int_{\color{red}{0}}^{+\infty}\frac{\rho\cos(\rho^2)}{\exp(\rho^2)}\,d\rho=\pi\int_{0}^{+\infty}\cos(z)e^{-z}\,dz = \color{red}{\frac{\pi}{2}}. $$