# Deriving some Laplace transforms

1. May 16, 2010

### Jamin2112

1. The problem statement, all variables and given/known data

Find the Laplace transform of each of the following functions:

....

2. f(t)=eatcos(bt)

3. f(t)=tn, were n is a positive integer

2. Relevant equations

As you well know, taking the Laplace of f(t) means ∫f(t)e-stdt from 0 to ∞

3. The attempt at a solution

These problems are tripping me up, since integration by parts goes on forever.

∫eatcos(bt)e-stdt =
∫et(a-s)cos(bt)dt =

I suppose I should call et(a-s) "dv" (?)
----> v = t(a-s)/(a-s)
----> u = cos(bt)
----> du = -bsin(bt) dt

Still, I don't see where this gets me. Help, please!

2. May 16, 2010

### hgfalling

Do the integration by parts again, and then collect the integral you are interested in on one side.

This is like the following simpler integral:

$$\int e^x \sin x dx$$
$$\int e^x \sin x dx = -e^x \cos x - \int -e^x \cos x dx$$ (one IBP)
$$\int e^x \sin x dx = -e^x \cos x + e^x \sin x - \int e^x \sin x dx$$ (second IBP)
$$2 \int e^x \sin x dx = e^x \sin x - e^x \cos x$$ (collect like terms)
etc...

3. May 16, 2010

### Jamin2112

In the case of this particular problem, what integral am I interested in?

4. May 16, 2010

### hgfalling

Well, you want

$$\int_{0}^{\infty} e^{t(a-s)} \cos (bt) dt$$

which is related, but with, you know, constants and things.