Laplace Transform of e^(-t)cos2t u(t-1)

[Moneer81]

Homework Statement

What is the laplace transform of e^(-t) cos 2t u(t-1)

Homework Equations

definition of Laplace transform: LT of f(t) = integral of f(t)e^-st dt, where limits of integration are from 0 to infinity

The Attempt at a Solution

since I have u(t-1) then do I just change the limits of integration to go from 1 to infinity instead?


then I guess what is the fastest way to evaluate the resulting integral:
integral of e^-(s+1)t cos 2t dt ?

 

[sprint]

my hint is that the step function has a big influence on the limits of integration since the step function is zero to the left side of when the step function is one.

and then...maybe I am not sure, you could use a trigonometry identity

 

[Moneer81]

OK so I was right in changing the lower limit of integration from 0 to 1?

As far as the integral goes, for a similar problem in a book I was reading they ended up with an expression that was somehow obtained from the quotient rule of derivatives...

 

[sprint]

u(t-1)

take t - 1 = 0 -----> t = 1

but I am not sure if changing the limits of integration forces you to change the "t's" in the function.

ex. cos 2(t-1) and e^-((s+1)(t-1))