I know that the Dirac Delta "function" $\delta(x)=\begin{cases}\infty&x=0\\0&x\neq0\end{cases},\int_{\Bbb{R}}\delta(x)\,\mathrm{d}x=1$ is supposedly a distribution, not a genuine function, (although I know nothing about distributions and what that really means). I also know that it has various integral representations, of which the one I know best I will present here:
$$\delta(\omega)=\frac{1}{2\pi}\int_{\Bbb{R}}\exp(it\cdot\omega)\,\rm{dt}$$
Formally I assume this integral does not actually exist for $\omega\neq0$, since $\lim_{n\to\pm\infty}\exp(in\cdot\omega)$ does not exist due to oscillating behaviour, so this representation raises some concerns for me. This representation is used to present answers to Fourier transforms, and that integral (which I do not believe exists) is "calculated" and whisked under the rug of Dirac Delta. Is this a correct way to define the distribution?
I am aware that in physics/engineering, it is common to heuristically use $\delta$ like this, with such representations, and treat it as a valid function, since the real world is too complex to treat everything with absolute rigour, but in the world of rigorous mathematics, I ask: is this correct?
For example:
$$\begin{align}\mathcal{F}(\cos)(\omega)&=\frac{1}{\sqrt{2\pi}}\int_{\Bbb{R}}\cos(t)\exp(it\cdot\omega)\,\mathrm{dt}\\&=\frac{1}{2\sqrt{2\pi}}\int_{\Bbb{R}}\exp(it(\omega+1))+\exp(it(\omega-1))\,\mathrm{dt}\\&\overset{\color{red}*}=\sqrt{\pi/2}\cdot(\delta(\omega+1)+\delta(\omega-1))\end{align}$$
Which correctly suggests that cosine consists of pure tones at the radial frequencies $\omega=\pm1$ (although the multiplication by $\sqrt{\pi/2}$ means what, exactly?)
Is this, namely the step denoted by $\color{red}*$, formal (in the context of distributions)? Or is this just some standard physicist's heuristic?
Many thanks! Note that I really do not know any distribution theory, but am just very curious at where the line between rigorous Dirac/Fourier and heuristic Dirac/Fourier is drawn!