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# Integration by Parts / Laplace Transform watch

1. I'm not even sure that what I'm about to ask is possible. Just a pre-warning.

Because I live such a rivoting life I've spent some time this Friday evening trying to find the Laplace Transform of , where n is a positive integer constant.

Why am I doing this? Well, the transform of t is easy to find, requiring just one iteration of IBP. t^2 takes a bit more effort, but is also relatively easy. Anything else is somewhat overkill. Plus I'm bored.

Now, the Laplace Transform of a function f(t) is given by

i.e. for this function, we're considering

IBP:

Letting

so and

so that

which simplifies down to

Which also requires integration by parts. Using the same steps as above, we come to

I would imagine, but haven't checked that the next stage would come out as

Obviously this continues n times.

My question is this - how do you go about actually evaluating this integral? It's probably something blindingly obvious but my brain has stopped working. Or is it just not doable using elementary functions?
2. Well let . You've found that , and so you can apply this formula recursively (putting and so on) until you get , but is something you can evaluate.
3. (Original post by nuodai)
Well let . You've found that , and so you can apply this formula recursively (putting and so on) until you get , but is something you can evaluate.
Not sure if it's a typo on your part, or obtuseness on mine, but how did a factor of e^-st get into the RHS? Surely that's contained within .

EDIT: lol, speaking of typos... *sigh* [/irony]
4. (Original post by EEngWillow)
Not sure it it's a typo on your part, or obtuseness on mine, but how did a factor of e^-st get into the RHS? Surely that's contained within .
Woops yeah, that was definitely a typo.
5. (Original post by nuodai)
Woops yeah, that was definitely a typo.
Anyway, just using this recursively is giving out the results I'd expect. Thanks! :-)
(I knew it was something obvious! xD)
6. You might also want to look up the gamma function.
7. (Original post by DFranklin)
You might also want to look up the gamma function.
Thanks for the suggestion. Will give it some looking at later, right now I have a cake to make.
(Does the gamma function make this easier or more difficult than nuodai's suggestion?)
8. The gamma function is essentially the same as the integral you're wanting to find. (So you'll see lots of possible approaches for evaluating it.

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