Laplace Transform of Sine/Proof 2

Theorem
Let $\sin$ be the real sine.

Let $\mathcal L$ be the Laplace Transform.

Then:


 * $\displaystyle \mathcal L \left\{{\sin at}\right\} = \frac a {s^2+a^2}$

where $a \in \R$ is constant, and $\operatorname{Re}\left({s}\right) > a$.

Proof
Also:

So: