Question

What is the Laplace Transform of (e^-t)sin(t)

Answer 1

According to my good old CRC Standard Math Tables, it should be something like\[\frac{1}{(s+1)^2+1}\]

Answer 2

I need a detailed algebraic solution. I can get this with the table of transforms, however, I cannot seem to solve it algebraically.

Answer 3

@NChaston you could you the integral method to evaluate this
\[L(f(t)=\int_0 ^{\infty} f(t) e^{-st} dt\]

Answer 4

Yes, and then I have to integrate by parts but I cannot seem to get the answer which is shown in the table and was provided to me above.

Answer 5

It's a very hairy integration. I think I should get someone to show me one on one in person.

Answer 6

No need for by parts
\[\int_0 ^{\infty} e^{-t} \sin t\ e^{-st} dt\]
Replace sin t by
\[ \sin t= \frac{e^{jt}-e^{-jt}}{2}\]
now you can solve it, without by parts

Answer 7

Really? I didn't know that. That's a pretty handy trick, I'd say. I'll try that out right away, thanks a lot. Are there other situations where I can use that instead of using integration by parts to evaluate?

Answer 8

e^-t <-- shifts frequency by +1
sin at -> a/(s^2 + a^2) <-- put a = 1, and shift s by s+1

Answer 9

It seems as though you've made a hyperbolic substitution. Is that correct? Why does that work?

Answer 10

easiest way ...get yourself a Laplace transform table.

Answer 11

Haha yeah we have the table. We are required to solve a few using first principals, however.

Answer 12

for sin t and cos t you can use this
it's complex, sin and cos are defined like this, so it'll work surely

Answer 13

...unfortunately

Answer 14

I got one more thing:)
you could replace sin t as
\[\sin t= Im \ ( e^{jt})\]
Im= Imaginary part of
this is easier than the previous method

Answer 15

normally i take this as formula
http://www.wolframalpha.com/input/?i=integrate+sin%28at%29e^%28bt%29

Answer 16

Im(e^at) <--- this is faster method.

Answer 17

just put b = -(s+1)

Answer 18

So sin(t)=Im (e^jt) I've never seen this Imaginary part of thing. I am familiar with complex numbers and whatnot.

Answer 19

What would the solution look like using this imaginary method, if you don't mind my asking?

Answer 20

evaluate the integral putting e^(ibt) as usual ... but in the end ... just take the imaginary part.

Answer 21

Okay, my only question is that before I wasn't evaluating the integral with a complex portion, only e^-bt.

Answer 22

change sin into

Answer 23

yes:)
put the imaginary part outside the integral
\[Im\ (\int_0 ^{\infty} e^{-t} e^{jt} e^{-st}dt)\]

Answer 24

after evaluating you'll get a real + an imaginary term
imaginary term is your solution

Answer 25

just evaluate that ... isn't it easier. At last simplify it the integral in the form a+ib
and just 'b' is your answer.