How to find the inverse laplace transform of [F(s)/s]^n

Question

Let $F(s)=\mathcal{L}\{f(t)\}$, we have $\frac{F(s)}{s}=\mathcal{L}\{\int_o^tf(x)dx\}$. How to find $\mathcal{L}^{-1}\left\{\left(\frac{F(s)}{s}\right)^n\right\},\text{ for}~ n\in \mathbb{N} $

Answer 1

A related problem. Here is a start for the case $n=2$,

Let

$$ g(t)=\int_{0}^{t} f(x)dx .$$

Now, recalling the fact

$$ \mathcal{L}(g*h) = \mathcal{L}(g) \mathcal{L}(h), $$

we have

$$ \mathcal{L}^{-1}\left\{\left(\frac{F(s)}{s}\right)^2\right\}=(g*g)(t)=\int_{0}^{t}g(\tau)g(t-\tau)d\tau .$$

You can simplify the above integral by interchanging the order of integration and see what you get. Try to generalize this answer.