In a lot of my engineering textbooks the order of integration on improper integrals is mindlessly swapped in proofs regarding properties of the Fourier Transform. This has always bothered me: when I look up when this can be done, I find theorems which heavily restrict the conditions on the input functions, for example requiring absolute integrability and integration over a definite area. At the same time, functions like $\frac{\sin(t)}{t}$, $\cos(t)$ and constants are used in Fourier Transforms and it is assumed that these properties apply, even though these functions are not absolutely integrable. I have found the unfamiliar term "tempered distributions" a lot which supposedly helps this against this problem, but I am really not well-versed in distribution theory. My question is: are the operations made by these engineers allowed using the constraints that the Fourier Transforms of the input functions $f(t), h(t)$ exist, even including the delta distributions present in some of these transforms?
I do not have a background in mathematics, thus engineering mathematics are the only tool on which I can really rely here. I have a few examples of steps made below:
$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(q)h(t-q)e^{-i2\pi F t}dqdt = \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(q)h(t-q)e^{-i2\pi F t}dtdq$
$\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(t - q)e^{i2\pi F q}dFdq = \int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(t - q)e^{i2\pi F q}dqdF$