Integration by parts and Laplace Transforms

[damo03]

Hi All,

This is not a homework question, I am just trying to be come quicker at integrating by parts, when performing Laplace Transforms.

My textbook gives a basic example for performing the Laplace Transform of the variable t, to the transformed variable of s for the

equation: f(t)=t^2

It then provides this working for the solution:

Now, I do not understand how they have "evaluated the integral on the right hand side of the equation". The book provides no "list of integrals" and I have NO idea how they got this within a few lines? It seems as though there is some sort of almost quadratic they use to speed things up but I can't make out the rule.

I can do integration by parts, which takes a while, or I can use the method (example 9) a the bottom of this page

http://tutorial.math.lamar.edu/Classes/CalcII/IntegrationByParts.aspx

which is much quicker. But if someone could please tell me how the textbook does it in so few lines that would be much appreciated.

Thanks

 

[damo03]

sorry mods, I should have posted this in the "homework and coursework section"...

 

[chiro]

I would have used integration by parts myself to get the answer. Maybe they just did this and got straight to the answer because they assumed that it was obvious (unfortunately many authors do that).

 

[homology]

I second the motion! Yeah I'd use integration by parts as well. There's a great little quote which is a footnote in Griffiths intro quantum book regarding integration by parts which I've found helps speed it up,

"Under the integral sign, then, you can peel a derivative off one factor in a product and slap it onto the other one - it'll cost you a minus sign, and you'll pick up a boundary term."

(bottom of page 15)

So something like \int t^2 e^{-st}dt=t^2(\frac{-1}{s}e^{-st})|_{stuff}-\int 2t (\frac{-1}{s}e^{-st}) dt

where I didn't simplify anything on purpose. Try doing integration by parts in your head, then do it out the long way and compare.

The being said you can find Laplace transform tables all over the place.

 

[jhooper3581]

Why would you put this in homework section if you forced yourself to admit that this is not a homework problem?

See http://en.wikipedia.org/wiki/Integration_by_parts" .