Laplace Transform of $ f(t)=ta^t $ and $ f(t)=t\sin(at) $

Question

How do I solve the Laplace transform of $ f(t)=ta^t $ and $ f(t)=t\sin(at) $?

Has some property that can help me solve or I can just revolve using the definition?

I can not solve... Help me please. Thanks!

Answer 1

The second is directly from the tables so I can't quite hint you on how to do it unless you want to prove it. $$\mathscr{L}_t\left [t\sin (at)\right ](s)=\dfrac{2as}{(a^2+s^2)^2}$$

But I can hint you about the first one. If $F(s)\colon =\mathscr{L}_t[f(t)](s)$ then the Laplace transform of $\mathrm{e}^{ct}f(t)$ is $F(s-c)$. All you gotta do is find what your $f(t),c$ are in your problem.