Hello I have the following function in the Laplace domain:

\(\frac{1}{s \times (s+1)}\)

For which I get the inverse Laplace transform: \(1-e^{-t}\) but looking at the solution they claim it's \((1-e^{-t})\times u(t)\) where u(t) is the unit step function. I don't understand why that is?

Also for \(\frac{1}{s^2 \times (s+1)}\) i use the method of partial fraction expansion and get \(-u(t)+tu(t)+e^{-t}\) but they claim it's \((t-1+e^{-t})u(t)\).

 

The Laplace Transform you are working with is the single-sided Laplace Transform, meaning that the function you are taking the transform of has to be identically zero for t<0. Thus, when you take the inverse transform, you have to get as a result a function that is identically zero for t<0.