## Laplace transform, need help with integral

1. The problem statement, all variables and given/known data
I'm trying to solve a definite Laplace transform. The function is defined as $$f(t) = sin(t)$$ over the interval $$[0, \pi]$$

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

3. 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.

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 Mentor The trick is when you get back to the sine, move the integral to the LHS, so you end up with $$(\textrm{some stuff}) \int_0^\pi e^{-st}\sin t\, dt = \textrm{other stuff}$$