We want F(s) = L[A*(1 – e - btu(t)] = A* L (u(t) – e - btu(t))
F(s) = A*[ L(u(t)) – L(u(t)*e - bt)] because the Laplace transform is linear.
We know that L[u(t)] = 1/s
And L[u(t)*e – bt] = 1/(s+b)
Thus we get F(s) = A*(1/s – 1/(s+b))
Which is: