Laplace Transform of t cos(2t)

  • #1

Homework Statement


Find the Laplace Transform from the definition of f(t) = tcos(2t)


Homework Equations


[tex]\int e^-^s^ttcos(2t)dt[/tex]


The Attempt at a Solution


I started by doing parts
[tex]u = t[/tex]
[tex]du = dt[/tex]

[tex]dv = cos(2t)e^-^s^t dt[/tex]

but I get stuck on v and as far as I can tell doing parts on v won't help because neither term will be reduced when differentiated.

I'm looking for some direction on how to solve this integral.
Thanks!
 

Answers and Replies

  • #2
vela
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You might try splitting it up differently. Try

[tex]\begin{align*}
u & = t e^{-st} \\
dv & = \cos 2t
\end{align*}
[/tex]

You'll have to integrate by parts at least twice.
 
  • #3
Thanks. I was able to figure it out after starting with it split how you suggested.
 
  • #4
42
0
A sneaky way is to use the fact that [itex] L[t f(t) ] = - F^{\prime}(s) [/itex]. You can use the fact that [itex] L[ \cos (2t) ] = \frac{s}{s^2 + 4} [/itex] for s>0. This way you are taking a derivative instead of integrating!
 
Last edited:

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