laplace transform for f(t) = e^at

[jamster114]

Can anyone help me with this?

I have to derive a formula for the laplace transform for f(t) = e^(at) and use the formula derived to find the laplace transform for f(t) = e^(-2t)

i've been trying to find out what exactly the laplace transform is and im having no luck.

 

[jamster114]

I guess a laplace transform is when F(s) = L{f(t)} = integral (e^(-st)*f(t))dt from
0 to infinity.

so i plug in e^(at) for f(t) , solve the integral using -s and a as constants , not exactly sure what to do now.

 

[galactus]

\(\displaystyle \L\\\int_{0}^{\infty}e^{(a-s)t}dt=\frac{1}{s-a}\)

Now, sub in a=-2:

\(\displaystyle \L\\\frac{1}{s+2}\)