Help Proving a Complex Laplace Transform

[EaglesFan7]

Homework Statement

Hi, for a homework problem I have to prove this LaPlace transform, but I've never seen anything like this because it's in terms of x and t. I'm assuming x can just be left as constant but I'm not sure how to account for the exponent and the denominator. This definitely isn't something you can find on a table but I tried to break it down.

Relevant Equations

L(f(t)) = limit T ---> infinity of integral from 0 to T of e^(-st) * f(t)dt

So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd appreciate any help, thank you guys.

 

[Mark44]

Problem Statement: Hi, for a homework problem I have to prove this LaPlace transform, but I've never seen anything like this because it's in terms of x and t. I'm assuming x can just be left as constant but I'm not sure how to account for the exponent and the denominator. This definitely isn't something you can find on a table but I tried to break it down.
Relevant Equations: L(f(t)) = limit T ---> infinity of integral from 0 to T of e^(-st) * f(t)dt

So I could just try using the definition by taking the limit as T goes to infinity of ∫ from 0 to T of that entire function but that would be a mess. I tried breaking it down into separate pieces and seeing if I could use anything from the table but I honestly have no clue I'm really stuck. I'd appreciate any help, thank you guys.

What is there to prove? Your relevant equation is basically the definition of the Laplace transform of a function f, and you use definitions -- you don't prove them.

Here's your equation in a nicer format:
$$\mathcal L(f(t)) = \lim_{T \to \infty}\int_0^T e^{-st}f(t)dt$$
Click on the equation I wrote to see the LaTeX script I wrote.
Also, there's a link to our tutorial on LaTeX at the bottom left of the pane.

 

[HallsofIvy]

Yes, here x is just an arbitrary constant.