Derivatives and Integrals of Laplace Transforms

We've seen what the transform of a derivative or an integral is. Now we're going to examine what the inverse transform is when you derive or integrate the transform of some function. This is one of those tricks that you won't use often but will save you a whole heap of work and frustration when you get a problem if fits in well with.

\(\Laplace\{tf(t)\} = -F'(s) \implies \Laplace^{-1}\{F'(s)\} = -tf(t)\)

\(\Laplace\{\frac{f(t)}{t}\} = \int_s^\infty F(\tilde{s})d\tilde{s} \implies \Laplace^{-1}\{\int_s^\infty F(\tilde{s})d\tilde{s}\} = \frac{f(t)}{t} \)