ok can someone tell me why my work is wrong, it seems the book has what i have up to this point, except the have negative signs where i have positive ones...
thanks in advance...
Since it seems the book wanted you to use the convolution theorem (A very powerful tool) lets give it a go. This can be inverted by taking three easy integrals.
$\displaystyle \mathcal{L}^{-1}\left\{ F(s)G(s)\right\}=\int_{0}^{t}f(t-\tau)g(\tau)d\tau$
So start with
$\displaystyle \mathcal{L}^{-1}\left\{\frac{1}{s}\cdot \frac{1}{s-1} \right\}=\int_{0}^{t}1\cdot e^{\tau}d\tau=e^{t}-1$
Now use the convolution theorem again to get
$\displaystyle \mathcal{L}^{-1}\left\{\frac{1}{s}\left(\frac{1}{s}\cdot \frac{1}{s-1}\right) \right\}=\int_{0}^{t}1\cdot (e^{\tau}-1)d\tau=e^{t}-t-1$
And one more time to get
$\displaystyle \mathcal{L}^{-1}\left\{\frac{1}{s}\left(\frac{1}{s^2}\cdot \frac{1}{s-1}\right) \right\}=\int_{0}^{t}1\cdot (e^{\tau}-t-1)d\tau=e^{t}-\frac{t}{2}-t-1$