Integration by parts and Laplace Transforms

In summary, the conversation discusses the use of integration by parts and Laplace Transform to solve equations. The person is seeking help in understanding a specific example provided in their textbook and is unsure of how the integral on the right hand side of the equation was evaluated. They mention a quote from a book that helps with integration by parts and suggest using a table to find Laplace transforms.
  • #1
damo03
7
0
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:

5486719801_3ec7b85467.jpg


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
 
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  • #2
sorry mods, I should have posted this in the "homework and coursework section"...
 
  • #3
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).
 
  • #4
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 [tex]\int t^2 e^{-st}dt=t^2(\frac{-1}{s}e^{-st})|_{stuff}-\int 2t (\frac{-1}{s}e^{-st}) dt[/tex]

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.
 
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  • #5
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" .
 
Last edited by a moderator:

1. What is integration by parts?

Integration by parts is a calculus technique used to evaluate integrals of products of functions. It involves breaking down a complex integral into simpler components and then using the product rule to integrate them. This method is particularly useful when dealing with integrals involving trigonometric, logarithmic, or exponential functions.

2. How is integration by parts applied?

To apply integration by parts, you need to identify the parts of the integral that can be differentiated and integrated separately. Then, you use the product rule to determine the new integral, which can be solved using integration tables or by using integration by parts again. This process is repeated until the integral can be evaluated.

3. What is the Laplace Transform?

The Laplace Transform is a mathematical tool used to solve differential equations. It transforms a time-domain function into a complex frequency-domain function, making it easier to analyze and solve differential equations that cannot be solved using traditional methods. It is widely used in engineering, physics, and mathematics.

4. How do you compute the Laplace Transform of a function?

The Laplace Transform of a function is computed by integrating the function multiplied by an exponential term. This exponential term depends on the variable of integration and the initial conditions of the function. The result is a complex function of the variable s, which represents frequency in the transformed domain.

5. What are the applications of Laplace Transforms?

Laplace Transforms have various applications in engineering, physics, and mathematics. They are used to solve differential equations in control systems, circuit analysis, and signal processing. They can also be used to solve problems in heat transfer, fluid mechanics, and quantum mechanics. In addition, Laplace Transforms have applications in probability and statistics, such as in the Laplace distribution and Laplace smoothing.

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