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

[schapman22]

Homework Statement

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

But this one I have no clue where to begin
∫₀^tf(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

∫₀^∞e^-stf(t)dt

∫₀^tf(t)dt transforms to F(s)/s

The Attempt at a Solution

I have it set up as
∫₀^∞[∫₀^tf(t)dt]e^-stdt

 

[schapman22]

Sorry I know it is difficult to read with the superscripts and subscripts. I didn't know a better way of displaying it.

 

[HallsofIvy]

Yes, that's correct. Now do the integral using integration by parts
Let "u" be $\int_0^t f(t)dt$. What is du?
Let "dv" be $e^{-st}dt$. What is v?

 

[QuarkCharmer]

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.

 

[schapman22]

Ok so du would be f(t)?
and v would be -1/s e^-st

 

[QuarkCharmer]

Ok so du would be f(t)?
and v would be -1/s e^-st

That's right!

 

[schapman22]

Im sorry but I am still having trouble with one. Can you help me with how to proceed?

 

[schapman22]

I have it written out as uv - ∫vdu, but I don't know how to go from there.

 

[sunjin09]

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

 

[schapman22]

thank you HallsofIvy, QuarkCharmer, and sunjin09.