Difficult integral: e^(-st) sin(at) cost(bt)

[Trenters4325]

How do you do the integral of

e^(-s*t)*sin(a*t)*cost(b*t)?

I tried parts but that only made it more complicated.

 

[galactus]

Here's what Maple gave me:

 

[Trenters4325]

So it would probably take forever to do by hand, right?

 

[galactus]

Someone else may have a clever way, but I'd use technology for this one.

That's what things like TI-89's and Maple's are for.

If you must, you could start with maybe changing

\(\displaystyle \L\\e^{-st}sin(at)cos(bt)\) into

\(\displaystyle \L\\\frac{1}{2}\int{e^{-st}sin(at+bt)}dt+\frac{1}{2}\int{sin(at-bt)}dt\)

Anyway you look at it, it ain't gonna be 'purdy'.