Find the laplace transform by intergration
f(t)=tcosh(3t)

Integrate by parts, setting




To evaluate , integrate by parts again, this time setting






Integrate by parts yet again, with









So we have





We already have the antiderivative for the first term:




And we can easily find the remaining term's antiderivative by integrating by parts (for the last time!), or by simply exchanging with in the derivation of , so that we have




(The exchanging is permissible because and ; there are no alternating signs to account for.)

And so we conclude that