Solving Laplace Transform: L{texp(9t)sin(2t)}

[Pete_01]

Homework Statement

Find the Laplace transform of L{texp(9t)sin(2t)}

Homework Equations

(-1)^n F^n (s)

The Attempt at a Solution

I am not sure how to start it. The exp(9t) term throws me off. Usually I would just do (-1) F'(s) where F(s) is the laplace of sin(2t). But do I take the derivative of exp(9t) in addition to sin(2t)? Multiply them together?
Thanks in advance!

 

[HallsofIvy]

You can start by using the definition of Laplace transform:
$$\int_0^\infty f(t)e^{-st}dt$$

Here, $f(t)= t e^{9t} sin(2t)$ so that integral is
$$\int_0^t te^{(9- s)t} sin(2t) dt$$

Integration by parts should do that. The obvious choice is u= t, $dv= e^{(9-s)t}sin(2t)dt$ and you will have to do another integration by parts to find v.