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Laplace Transform

Hi,

How I would go about deriving the laplace transform for L[integral of f(t) from zero to t]?

I'm aware of how you derive something like L[df/dt] using the standard definition of the Laplace transform, but I can't make sense of this particular derivation!

Any help would be much appreciated! Thanks!

Reply 1

DFranklin

I'm not clear; is the problem that you have a derivation you don't understand, or that you need to find a derivation and can't work out how to do it?

Reply 2

Bioluminescence1

Original post by DFranklin

I'm not clear; is the problem that you have a derivation you don't understand, or that you need to find a derivation and can't work out how to do it?

The actual question is 'Derive the following Laplace Transform L[integral f(t) from zero to t].

I'm unsure how to derive it, as you can't use the normal definition?

Sorry, I don't know how to use the fancy maths format on here!

Reply 3

atsruser

Original post by Bioluminescence1

Write

F(t)=\int_0^t f(u) \ du

then use integration by parts on the Laplace transform integral. And think about what

F(0)

is.

Reply 4

Bioluminescence1

Original post by atsruser

Write

F(t)=\int_0^t f(u) \ du

then use integration by parts on the Laplace transform integral. And think about what

F(0)

is.

Sorry I'm not sure I get what you mean, could you expand please? (:

Reply 5

RichE

Original post by Bioluminescence1

Sorry I'm not sure I get what you mean, could you expand please? (:

It may be more straightforward to define F as above, differentiate both sides and then apply the Laplace transform. (It's tantamount to doing the same thing in the end.)

Reply 6

Bioluminescence1

Original post by RichE

It may be more straightforward to define F as above, differentiate both sides and then apply the Laplace transform. (It's tantamount to doing the same thing in the end.)

Differentiate and not integrate? Why would you do that?

Reply 7

RichE

Original post by Bioluminescence1

Differentiate and not integrate? Why would you do that?

Did you try it? What did you get when you differentiated F(t) and then transformed your answer?

Reply 8

atsruser

Original post by Bioluminescence1

Sorry I'm not sure I get what you mean, could you expand please? (:

First, do you know what result that you're trying to get to? If not, it'll probably help to look it up. Then, given the expression I wrote above:

1. We have

F(t) = \int_0^t f(u) \ du

2. We have

\mathcal{L} \{F(t)\} = \int_0^\infty e^{-st} F(t) \ dt

3. What is

F'(t)

?

4. What is

F(0)

?

5. How does that suggest that you perform the IBP, given the result that you want?

Reply 9

DFranklin

Original post by Bioluminescence1

Differentiate and not integrate? Why would you do that?

Well, if you know the relationship between the Laplace transform of f(x) and the transform of f'(x), let

g(x) = \int_0^x f(t)\,dt

then since you know the relationship between the Laplace transform of g and that of g', and you know the laplace transform of g' is the laplace transform of f, you should be able to use this to find the laplace transform of g.