Wikipedia page here talks about suitable conditions for which the following formula holds good.
$$\int_{0}^{\infty} f(x) g(x) dx = \int_{0}^{\infty} (\mathcal{L}f) (s) . (\mathcal{L}^{-1} g) (s) ds$$
I am not able to understand the assumptions and conditions stated there. Can someone please show with an example the cases where this formula cannot be applied?