Verify the following results by direct integration

laplace of ( e^-t*sin2t)= 2/s^2+2s+5

Question

Verify the following results by direct integration

laplace of ( e^-t*sin2t)= 2/s^2+2s+5

hartnn · Answer

where are u stuck ? 
u realize u need to use product rule for integration there ....

anonymous · Answer

yes im pretty much stuck ? product rule... i dont think you diffrentiate.

anonymous · Answer

you mean integration by parts?

hartnn · Answer

yes, i meant 'by parts'

anonymous · Answer

working on it ....

anonymous · Answer

i think you use the quotient rule not product rule  .... any way i got  -1/5*e^ -t*sin2t+2cos2t ,is that ron the right lines?

hartnn · Answer

where did s go ? 
didn't u realize u need to integrate 
$$\int\limits_{0}^{\infty}e^{-st}e^{-t}\sin2tdt$$

anonymous · Answer

can you use the laplace tables or do you have to do it from first principles?

hartnn · Answer

write that as $e^{-t(s+1)}$ and then integrate by parts

hartnn · Answer

when u integrate by parts you should get $e^{-t(s+1)}[-(s+1)sin2t-2cos2t]/(s^2+2s+5)$

hartnn · Answer

tell me if u don't get that

calculusfunctions · Answer

First of all do you know Laplace transform? If not I can write up a very brief lesson to explain.

anonymous · Answer

Right i have the same ans as  hartnn , but isn't the overall ans =2/s^2+2s+5...?

hartnn · Answer

yeah, now put the limits, first upper limit, t= infinity , then lower limit, t=0
u'll get the required answer

anonymous · Answer

right i got it ! finally thank you :)

hartnn · Answer

welcome ^_^