A question on the Laplace Transform of $f(t)=t e^{at}\sin (bt)$

Question

I would like to solve the Laplace transform of the following function:

$$t \mapsto t e^{at}\sin (bt).$$

I know that $\mathscr{L}\left(\sin(bt)\right)=\dfrac{b}{s^2+b^2}$ and that you have to take the integral from $0$ to $\infty$ and multiply it by $e^{-st}$.

Answer 1

Hint. From standard properties (see this table) and from the given result one gets $$ \mathscr{L}\left(t \sin bt \right)(s)=-\left(\mathscr{L}\left( \sin bt \right)\right)'(s)=\frac{2bs}{(s^2+b^2)^2} $$ then using $$ \mathscr{L}\left(e^{at}f(t) \right)(s)=\left(\mathscr{L}f\right)(s-a) $$ gives finally

$$ \mathscr{L}\left(t e^{at}\sin (bt) \right)(s)=\frac{2b(s-a)}{((s-a)^2+b^2)^2}. $$