Inverse Laplace transform of s/(s + 1)

Question

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.

Answer 1

The main issue with the second ``solution" is the following:

We have that \begin{align} \mathcal{L}^{-1}\{sF(s)-f(0)\}(t) = f'(t) \end{align} but \begin{align} \mathcal{L}^{-1}\{sF(s)\}(t) \neq f'(t)+f(0) \end{align} since \begin{align} \mathcal{L}^{-1}\{1\}(t)= \delta(t). \end{align} This means that \begin{align} \mathcal{L}^{-1}\{sF(s)\}(t) = f'(t)+\mathcal{L}^{-1}\{f(0)\}= f'(t)+f(0)\delta(t). \end{align}

Answer 2

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