What Does From First Principles Mean in Calculating Laplace Transforms?

[NotStine]

Homework Statement

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

Homework Equations

sin²3t = 1/2(1-cost(6t))

L{SINat} = a/(s² + a²)

The Attempt at a Solution

I already have the 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.

 

[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.

 

[NotStine]

I see. That makes sense.

Thank you very much for your help.