Integration by parts (Laplace transform)

[Speedo]

First off, I hope these images show up - I don't have time to figure out this latex stuff atm, so it's easier just to throw the formulae together in openoffice.

I'm working on the Laplace Transform for
http://home.directus.net/jrc748/f.gif

Which is obviously
http://home.directus.net/jrc748/l1.gif

Now, in case somebody brings it up, I know that we can look up the answer in the table of basic Laplace transforms as
http://home.directus.net/jrc748/l2.gif

But if I could get off that easy, I wouldn't be asking this question, would I? ;)

My main problem is that I'm fairly rusty on integration, so I'm probably missing something small. I'd picked out
http://home.directus.net/jrc748/uv.gif
and was attempting the tabular method, but I'm not getting anywhere. It's looking to me like no matter what you pick for u or dv you'll be doing integration by parts forever, since you're stuck with e^t and sin.
Just need some hints to kick start the thought process.

 

[devious_]

The usual trick when it comes to integrating a product of an exponential and a trig function by parts is that you will get the original integral again. So for example if we let the original integral be I then we will reach a step that looks like:
I = (some stuff) + kI => I = (some stuff)/(1 - k)

Maybe this could help you.

 

[Benny]

I haven't actually written the working out but I would try...

$$e^{ - st} t\sin \left( {at} \right) = t\left( {e^{ - st} \sin \left( {at} \right)} \right)$$

The t is quite pesky so get rid of it by using parts, differentiate the t and integrate the exponential and sine product (*). You'll be left with some complicated product (which you don't need to deal with) plus or minus some integral involving products of exponentials and sines or cosines. That should be easy to integrate.

(*) integrating the product of an exponential and sine (or cosine) should be standard procedure considering the level that you're at.

 

[Speedo]

Yep, that's the ticket. I figured it was right under my nose.

 

[Galileo]

A shortcut that is also much easier is to use the complex exponential:
$$\int_0^\infty te^{-(s-ia)t}dt$$
Your integral is just the imaginary part of this. As a side bonus you also get the Laplace transform of $t\cos(at)$ by taking the real part.