I've been working on a problem, this one in particular:

where is equal to cosh bt

I attempt to take the Laplace transform:

=

=

However, this is where I'm stuck. I'm not sure what do when I have a non-negative "t" (cause if it's negative, I can assume it's going to equal zero as A goes towards infinity). Can anyone show me where I went wrong, or what step is next? Thank you!