Question

laplace transform of cos (2pi)t

Answer 1

We have to integrate? D:

Answer 2

\[\Large \int\limits_{0}^\infty e^{-st}\cos(2\pi t)dt\]

Answer 3

Come on, @epicChoi make your presence felt :P

Answer 4

no

Answer 5

i dont know

Answer 6

You weren't taught how to do Laplace Transforms? D:

Answer 7

its so fast when it taught to us

Answer 8

Oh... yes, that is a problem... tell you what, let's do it for the general case, where instead of \(2\pi\) we have an arbitrary constant...say, a. And then, when we're done, just replace the a with the constant you desire, in this case \(2\pi\)
\[\Large \int\limits_{0}^\infty e^{-st}\cos(2\pi t)dt\]

Answer 9

By the way, THAT is the definition of a Laplace transform... the Laplace transform of a function f(t) is given by
\[\Large \int\limits_{0}^\infty e^{-st}f(t)dt\]

Answer 10

So, let's do this.... :)
\[\Large \int\limits_{0}^\infty e^{-st}\cos(a t)dt\]

Answer 11

Can you integrate this?
Never mind the limits for now, just integrate...
\[\Large \int e^{-st}\cos(a t)dt\]

Answer 12

thanks for lecture

Answer 13

Sarcasm? -.-

Answer 14

no

Answer 15

im not good in english :)

Answer 16

Korean, then? I don't speak Korean :P

Answer 17

hehe

Answer 18

^lol that was random... or not :D

Answer 19

what is your nationality?

Answer 20

Wouldn't you like to know :D
You first :P

Answer 21

you first please.

Answer 22

We could haggle about who goes first... or we could solve this integral :D
\[\Large \int e^{-st}\cos(a t)dt\]

Answer 23

solve it hehe

Answer 24

But this time, I must insist that you take the first step... at least identify HOW we should integrate this...

Answer 25

by parts?

Answer 26

parts or by treating it as the real part of the following complex-valued integral:$$\int e^{(ia-s)t}\,dt$$

Answer 27

Good call :)
...
Yikes... this is going to be monstrous...
let \(\large u = e^{-st}\) and \(\large dv = \cos(at) dt \)

from here, it follows that \(\large du = -se^{-s t}dt\) and \(\Large v = \frac{\sin(at)}{a}\)

Answer 28

dont be so complicated -_-'

Answer 29

It's going to be complicated one way or another... this is like derivatives... at first, you want to kill yourself (hopefully not really) when you're forced to do it using limits, but eventually, it becomes second-nature due to several rules and useful shortcuts...

Answer 30

So... by parts...
\[\Large \int e^{-st}\cos(a t)dt= \frac{e^{-st}\sin(at)}{a}+\int \frac{se^{-st}\sin(at)}{a}dt\]

Answer 31

please kill me now

Answer 32

Gladly......
LOL what am I saying...

oh yeah, do this integral...
\[\Large \int e^{-st}\cos(a t)dt= \frac{e^{-st}\sin(at)}{a}+\color{red}{\int \frac{se^{-st}\sin(at)}{a}dt}\]

Answer 33

i will kill u ?

Answer 34

thanks for the lecture. i will leave for now. i will buy food for dinner.

Answer 35

@epicChoi notice \(s/a\) is constant w.r.t. \(t\) hence:$$\int\frac{se^{-st}\sin at}adt=\frac{s}a\int e^{-st}\sin at\,dt$$

Answer 36

Of all the times to do your 'magic' you pick when our OP has left? -.-

Answer 37

i just want some fried chicken

Answer 38

give me medal or i tell ur dad on u

Answer 39

\[\mathcal L\big\{\cos(at+b)\big\}=\frac{s\cos b-a\sin b}{s^2+a^2}\]

Answer 40

you do not have to integrate, you just have to look it up in a table of laplace transforms, and substitute

Answer 41

@UnkleRhaukus the whole point is to justify the table's answer :-p

Answer 42

well that is not clear in the question.

Answer 43

http://openstudy.com/study#/updates/5104cfd1e4b03186c3f99b4f

Answer 44

^oh well :)