# Thread: Laplace transform help

1. ## Laplace transform help

Hey guys, I need some help with this problem:
[tex]\int e^-^s^t cost dt lower limit is 0 and the upper limit is \infty [\math]

The problem I am running into is when I use integration by parts. I let u = cost and dv = [tex] e^-^s^[\math]

I am somehow getting stuck with an infinite integration by parts problem. Any help is greatly appreciated!!! Thank you!

Sorry, for some reason the latex is not working the way I want it to.......

2. Let

$\displaystyle I=\int_{0}^{\infty}e^{-st}\cos(t)\,dt.$

Let

$\displaystyle u=e^{-st},\quad du=-se^{-st}\,dt,\quad dv=\cos(t)\,dt,\quad v=\sin(t).$

We get

$\displaystyle I=e^{-st}\sin(t)\Big|_{0}^{\infty}+\int_{0}^{\infty}se^{-st}\sin(t)\,dt=s\int_{0}^{\infty}e^{-st}\sin(t)\,dt.$

Next, we let

$\displaystyle u=e^{-st},\quad du=-se^{-st}\,dt,\quad dv=\sin(t)\,dt,\quad v=-\cos(t).$

We get

$\displaystyle I=s\left[-e^{-st}\cos(t)\Big|_{0}^{\infty}-s\int_{0}^{\infty}e^{-st}\cos(t)\,dt\right]=s\left[1-sI\right].$

It is that last step there, recognizing that we've arrived back at I, that is the crucial one. You now have an equation for I. Solve for it.

3. I think we're back to [TEX ] ... [/TE X]

4. Originally Posted by Jeonsah Hey guys, I need some help with this problem:
[tex]\int e^-^s^t cost dt lower limit is 0 and the upper limit is \infty [\math]

The problem I am running into is when I use integration by parts. I let u = cost and dv = [tex] e^-^s^[\math]

I am somehow getting stuck with an infinite integration by parts problem. Any help is greatly appreciated!!! Thank you!

Sorry, for some reason the latex is not working the way I want it to.......
Don't use integration by parts! Either use the complex exponential form for $\displaystyle \cos (t)$:

$\displaystyle \cos(t) = \frac{e^{it}+e^{-it}}{2}$

or write your integral as:

$\displaystyle \int_0^{\infty}e^{-st}\cos(t)\; dt= \text{re}\left(\int_0^{\infty}e^{-st}e^{it}\; dt \right)$

CB

5. Thanks guys, i figured it out with integration by parts. Ima make another topic with a couple new questions with the laplace transform.

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