I'm trying to invert the following equation back into the time domain
$$\frac{c}{s^{2}} \bigg(2-e^{-a\sqrt{s}}+e^{a\sqrt{s}}\bigg)$$ where ${a}$ and ${c}$ are positive constants.
I'm ok with solving the first two terms within the big brackets but I don't know how to deal with $\frac{e^{a\sqrt{s}}}{s^2}$. Are there any suggestions about how to tackle it?
Hint: Note that the convolution of the functions $f$ and $g$ is given by $$f\ast g=\int_{0}^{t}f(\tau)g(t-\tau)d\tau$$ Also, we can see that $$\mathscr{L}^{-1}\left\{F(s)G(s)\right\}=f\ast g$$ Now, note that $$\mathscr{L}^{-1}\left\{\frac{\exp(a\sqrt{s}}{s^{2}}\right\}=\mathscr{L}^{-1}\left\{\frac{1}{s^{2}}\cdot \exp(a\sqrt{s})\right\}=f(t)\ast g(t)$$ where $F(s)=\frac{1}{s^{2}}$ and $G(s)=\exp(a\sqrt{s})$.