# Laplace transform, need help with integral

## 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.

## Answers and Replies

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