laplaceneed

1. (Laplace)Need explainations

hey guys need some explanation on these: the red arrows indicates the part where i do not understand how the eqn derive to the other eqn. (see picture 3) And maybe if you can also explain the "pattern" of the t>1, 1<t<2, 2<t<4 ...you know what i mean,if the question changes. i am having a...