# Laplace from first principles

1. Nov 5, 2008

### NotStine

1. The problem statement, all variables and given/known data

Find, from first principles, the Laplace transform of f(t) = sin2(3t)

2. Relevant equations

sin23t = 1/2(1-cost(6t))

L{SINat} = a/(s2 + a2)

3. The attempt at a solution

s/2(s2 + 36)

What I want to ask is what does the first principles part mean? Is that asking me to integrate or look at the table of transforms? I don't quite understand what method I'm suppose to use for the transformation. Any light on this topic would be heavily appreciated.

Last edited: Nov 5, 2008
2. Nov 5, 2008

### HallsofIvy

The Laplace transform of a function, f(x) is defined as
$$L(f)(s)= \int_0^\infty e^{-sx}f(x)dx$$
"From first principles" means using that formula. I recommend integrating by parts.

3. Nov 5, 2008

### NotStine

I see. That makes sense.

Thank you very much for your help.