Laplace Transform of Sine/Proof 4

Proof
By definition of the Laplace transform:


 * $\displaystyle \laptrans {\sin at} = \int_0^{\to +\infty} e^{-s t} \sin a t \rd t$

From Integration by Parts:


 * $\displaystyle \int f g' \rd t = f g - \int f' g \rd t$

Here:

So:

Consider:


 * $\displaystyle \int e^{-s t} \cos a t \rd t$

Again, using Integration by Parts:


 * $\displaystyle \int h j\,' \rd t = h j - \int h' j \rd t$

Here:

So:

Substituting this into $(1)$:

Evaluating at $t = 0$ and $t \to +\infty$: