Sorry in advance if this question is a bit off topic. Here when i use Laplace Transform or Fourier Transform (these the only integral transforms I’m familiar with.) to solve Differential Equations i don’t see there’s a benefit to use the inversion formula and i think using table of Laplace/Fourier transform and partial fraction decomposition is easier and faster. Of course we can create our own tables by looking for the general integral transform of functions.
So what’s the point for having the inversion formula? Have been reading how to use the inversion formula especially about Bromwich Integral and it looks really complicated and needs some kind of contour analysis and such. Hope you don’t mind letting me know what are the benefits of having this formula at an advanced level.