Laplace transform of a piecewise function

[Feodalherren]

Homework Statement

f(t) = e^t when 0≤t<1
and 0 when t≥1

Homework Equations

Laplace transformations

The Attempt at a Solution

so the Laplace integral becomesfrom 0 to 1 ∫e^(st^2)dt + 0

how do I integrate this?

 

[Rellek]

Clarification:

Laplace transform or Lagrange Transform...?

 

[Feodalherren]

My bad, Laplace :).

 

[Rellek]

Alright, let's do this:

From 0 to 1 we have one function, and from 1 onward we have another. Split up our integral as so:

$$\int_0^1 e^{-st} e^{t}dt + \int_{1}^{\infty} e^{-st}(0)dt \implies \int_0^1 e^{t(1-s)}dt$$

 

[Feodalherren]

Wait a second, doesn't the part that goes from 1 to +infinity get canceled out because the integral becomes

∫ e^(-st) (0) dt = 0

?

 

[Rellek]

Wait a second, doesn't the part that goes from 1 to +infinity get canceled out because the integral becomes

∫ e^(-st) (0) dt = 0

?

Excuse my reading comprehension, I thought it said f(t) = 1. Corrected.

 

[Feodalherren]

Ah now I see what I did wrong! DUH! Such a stupid mistake.

Thank you sir!

 

[SteamKing]

Homework Statement

f(t) = e^t when 0≤t<1
and 0 when t≥1

Homework Equations

Laplace transformations

The Attempt at a Solution

so the Laplace integral becomesfrom 0 to 1 ∫e^(st^2)dt + 0

how do I integrate this?

How did you get an integrand of e^st2 ?

Remember, e^x ⋅ e^y = e^{(x + y)}, not e^xy

 

[Feodalherren]

I was just asking myself the same thing. I think I need to take a break. I've been doing math since 7.30 this morning. It's 1.30 pm now :).