Laplace transform, with integral

[arenaninja]

Homework Statement

I'm trying to solve a definite Laplace transform. The function is defined as $$f(t) = sin(t)$$ over the interval $$[0, \pi]$$

Homework Equations

The integrate to evaluate is:
$$\int_0^\pi e^{-st} sin(t) dt$$

The Attempt at a Solution

To evaluate, first use integration by parts (IBP).
$$\begin{matrix} u = sin(t) \quad dV = e^{-st} dt\\ du = cos(t) dt \quad V = \frac{e^{-st}}{-s} \end{matrix}$$

$$\int_0^\pi e^{-st} sin(t) dt = \frac{sin(t) e^{-st}}{-s} + \frac{1}{s} \int_0^\pi e^{-st} cos(t) dt$$
However, I'm stuck here. I can try to keep evaluating by parts, but it looks to me like I'm stuck in a loop. Integrating by parts will alternate me between sine and cosine, and the only thing that will change will be the increasing power for the "s" in the denominator.

Any help is greatly appreciated.

 

[vela]

The trick is when you get back to the sine, move the integral to the LHS, so you end up with

$$(\textrm{some stuff}) \int_0^\pi e^{-st}\sin t\, dt = \textrm{other stuff}$$