Transform of a convolution problem

Question

Find L^-1 {1/((s^3)*(s-1))} as a convolution and then evaluate the convolution

Answer 1

A Table Of Laplace Transforms going from f(s) to F(t) shows (1/s³) transformed to t²/2! and 1/(s-1) transformed to e^(1)t or e^t.

Then obtain, by Convolution, L^-1{1/(s³(s-1)} equal to ∫_{(from u=0 to u=t)}(u²/2!)e^(t-u)du.

Rewrite ∫_{(from u=0 to u=t)}(u²/2!)e^(t-u)du as (e^t/2!)∫_{(from u=0 to u=t)}u²e^-udu.

Integration By Parts will give (e^t/2!)∫_{(from u=0 to u=t)}u²e^-udu as (e^t/2!)[-u²e^-u −2ue^-u − 2e^-u|_{(from u=0 to u=t)}] which equals (e^t/2)[-t²e^-t − 2te^-t − 2e^-t + 2] or -t²/2 − t − 1 + e^t.