Laplace of t.e^at.sinat

[R.K.4.7]

Specifically I can't calculate the Laplace of questions like (at).e^(at).Sin(at)
I know there is formula for L{sint.e^t}
But I can't seem to find some method for the question I previously asked.
I'm also attaching a picture, check out the Q11 & Q12
They have a similar components when simplified...

 

[HallsofIvy]

The definition of "Laplace Transform" for function f is $\displaystyle L(f)= \int_0^\infty f(t)e^{-st}dt$. Surely that is in your textbook or notes?

So the Laplace Transform of $\displaystyle f(t)= ate^{at}sin(at)$ is $\displaystyle \int_0^\infty ate^{at}sin(t)e^{-st}dt= a\int_0^\infty t sin(at)e^{(a- s)t} dt$ and, as is typical for Laplace Transforms, that can be integrated by "integration by parts.

There are many different choices for "u" and "dv" but a good start is to take u= t, $\displaystyle dv= sin(at)e^{(a- s)t}$ which will eliminate the multiplied "t" from the integral.

 

[R.K.4.7]

Sorry man I didn't get the point here...
Do I have to just integrate the whole thing? Also what kind of eqn will I get ?

 

[HallsofIvy]

Yes, as I said, the definition of the "Laplace transform" is an integral and you need to do the integration to find the Laplace transform!