Laplace Transform for ∫0tf(t)dtA Guide to Solving Laplace Transforms

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Homework Help Overview

The discussion revolves around the application of Laplace transforms, specifically focusing on the integral of a function from 0 to t, represented as ∫0tf(t)dt. Participants are tasked with demonstrating the relationship between this integral and its Laplace transform, which is noted to be F(s)/s.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the setup of the integral and the application of integration by parts. There are questions about identifying the components of the integration by parts formula, particularly what "u" and "dv" should be. Some participants express confusion about how to proceed after setting up the integration by parts.

Discussion Status

Several participants have provided guidance on using integration by parts, with some confirming the identification of "du" and "v." However, there remains uncertainty among participants about the next steps in the process, indicating an ongoing exploration of the problem.

Contextual Notes

Participants have noted difficulties with the notation and the clarity of the problem setup, which may affect their understanding and progress. There is also a reference to the fundamental theorem of calculus as a potential aid in understanding the integral's behavior.

schapman22
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Homework Statement


We have been given a table of laplace transforms and have been asked to show them using the definition. ∫0e-stf(t)dt.

But this one I have no clue where to begin
0tf(t)dt the laplace transform of this is F(s)/s.
Can anyone tell me what to do with this one? Thank you in advance.


Homework Equations



0e-stf(t)dt

0tf(t)dt transforms to F(s)/s

The Attempt at a Solution



I have it set up as
0[∫0tf(t)dt]e-stdt
 
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Sorry I know it is difficult to read with the superscripts and subscripts. I didn't know a better way of displaying it.
 
Yes, that's correct. Now do the integral using integration by parts
Let "u" be [itex]\int_0^t f(t)dt[/itex]. What is du?
Let "dv" be [itex]e^{-st}dt[/itex]. What is v?
 
Check out the first part of the fundamental theorem of calculus, and think of that integral with the upper limit as t as some function.
 
Ok so du would be f(t)?
and v would be -1/s e-st
 
schapman22 said:
Ok so du would be f(t)?
and v would be -1/s e-st

That's right!
 
Im sorry but I am still having trouble with one. Can you help me with how to proceed?
 
I have it written out as uv - ∫vdu, but I don't know how to go from there.
 
Try to prove the simpler but related problem using integration by parts, then you'll have some clue:
L{g'(t)}=sG(s)-g(0).
Then let g(t)=∫ _{0 to t} f(τ)dτ
 
  • #10
thank you HallsofIvy, QuarkCharmer, and sunjin09.
 

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