I'm doing differential equations and specifically studying Laplace Transformations, where of course the Kernel is:

$K(s,t) = e^{-st}$

And the Laplace Transformation $\mathcal{L}$ of a function $f(t)$ is:

$\mathcal{L\{f(t)\}} = \int\_0^{\infty}K(s,t)f(t)\,dt$

So if I'm given $f(t) = t\cos(t)$, which gives me:

$\mathcal{L\{f(t)\}} = \int\_0^{\infty}e^{-st}t\cos(t)\,dt$

How do I go about integrating this? It's not nearly as simple as doing Integration By Parts since I have three different functions here $(e^{-st}, t, \cos(t))$. No ideas for $u$-substitution either. Is there a trig sub that I'm not seeing here?