Understanding the LaPlace Transformation for Solving Complex Integrals

[williamcarter]

Dear PF members.
I am requesting again your help as I keep struggling with the LaPlace transformation.
I have this exercise to do(please see below)

We know that L[f(t)]= integral from 0 to infinity of f(t)*e^(-st) dt
thus in our case, L[f(t)]= integral from 0 to infinity of sin(t)*e^(-st) dt

I tried doing it by parts twice, however I can't reach their answer.Please look below

Could you please show me how to solve it?

Thank you in advance.

 

[Fightfish]

Which part are you struggling with? Isn't your second attachment already the worked-out solution?

 

[Ray Vickson]

Dear PF members.
I am requesting again your help as I keep struggling with the LaPlace transformation.
I have this exercise to do(please see below)
View attachment 109363

We know that L[f(t)]= integral from 0 to infinity of f(t)*e^(-st) dt
thus in our case, L[f(t)]= integral from 0 to infinity of sin(t)*e^(-st) dt

I tried doing it by parts twice, however I can't reach their answer.Please look below
View attachment 109364
Could you please show me how to solve it?

Thank you in advance.

It is against PF rules for us to "show you how to solve it". At most, we can offer hints. However, since you have already done all the work, I can tell you that your final answer is correct.

Furthermore, the way you did it is one of the ways I would have done it; the other way would have been to write $\sin(t) = (e^{it} - e^{-it})/(2i)$ and integrate the two terms separately.

 

[williamcarter]

Thank you very much for confirming.