Another more useful expression of the f(t) is...
... where is the 'Haeviside step function'. Now You have to compute the LT of (1) 'term by term' in appropriate way...
The definition of the Laplace transform of a function is
For this particular function we have:
The first and third integral terms above are equal to , so the Laplace transform would be
Now, using integration by parts, I got the answer:
I kinda rushed through the integration, so check for yourself what the integral equates to.