Laplace Transform Let f be a function defined for t ≥ 0. Then the integral ℒ{f(t)} = [infinity] e−stf(t) dt 0 is said to be the Laplace transform of f, provided that the integral converges. to find ℒ{f(t)}. (Write your answer as a function of s.) f(t) = t2e−6t

The laplace transform is

Step-by-step explanation:

We will solve this problem by applying the laplace transform properties (their proofs are beyond the scope of this explanation).

Consider first the function f(t) = 1. By definition of the laplace transform, we have

when f(t) = 1 we get

We will apply the following properties: Define L(f) as applying the laplace transform

(this means, multiplying by an exponential corresponds to a shift in the s parameter of the transform of f)

(this is, multypling by is equivalent to taking the n-th derivative of the transform.

We are given the function . Since the transform of the constant function 1 is 1/s, by applying the first property we get

By applying the second property we get