Question

Let​ f(t) be a function on​ [0,

infinity[infinity]​).

The Laplace transform of f is the function F defined by the integral

​F(s)equals=Integral from 0 to infinity e Superscript negative st Baseline f left parenthesis t right parenthesis dt∫0[infinity]e−stf(t)dt.

Use this definition to determine the Laplace transform of the following function.

Answer 1

Laplace transform of f(t) = 2t² is given as

F(s) = (4/s³)

Defined for s > 0

Explanation:

The picture of the full question is shown in the first attached image to the solution I'll provide for this question.

Laplace transform of a function f(t) is given as

F(s) = ∫∞₀ f(t) e⁻ˢᵗ dt

For this question,

f(t) = 2t²

F(s) = ∫∞₀ (2t².e⁻ˢᵗ) dt

The integration is evaluated using integration by parts and it is presented in the second and third attached images to this solution.

The initial integration by parts led to an answer that still had an integral part that needed integration by parts to be evaluated again.

So, there are two uses of integration by parts in the solution provided.

At the end, the definite integral, ∫∞₀, is then introduced, and the definite integral is evaluated.

Note that e^(-∞) = 1/(e^∞) = 0

And e⁰ = 1.

The Laplace transform of 2t² was then obtained to be

F(s) = ∫∞₀ (2t².e⁻ˢᵗ) dt = (4/s³)

This Laplace transform is defined for s > 0

Hope this Helps!!!