Question

how to solve laplase f(t)=tcost

Answer 1

\[\frac{ s^{2}-1 }{ (s^{2}+1)^{2} }\]

Answer 2

if you have no idea what to do, use the definition of the transform. \[\huge L \left\{ t \cos t \right\}=\int\limits_{0}^{+\infty} (t \cos t) e^{-st}dt\]

Answer 3

this solution is lengthy, though.
another property of the transform is \[\large L \left\{ t^n f(t) \right\}=(-1)^n \frac{ d^n }{ ds^n }F(s)\] where \[\large F(s)=L \left\{ f(t) \right\}\]