Inverse Laplace transform of s/(s + 1)

I'm trying to understand what's the inverse Laplace transform of $\frac{s}{s+1}$.

I found this answer, which is quite clear and concludes that it's $δ(t)-e^{-t}$, which sounds right given the reasoning.

But I also found this proof, which calculate the inverse of $\frac{1}{s+1}$, then derives it and adds initial condition; with this proof, assuming $a=b=1$, the reverse transform is $1-e^{-t}$.

I'm not quite sure about what's different between these two conclusions, and I can't find anything wrong with any of them, so I was wondering how those solutions fits together, and if there is any error or approssimation in those proofs.

Solution 1:

Let write $$ \frac{s}{s+1}=1-\frac{1}{s+1} $$ The ILT is thus $\delta(t)-e^{-t}$.