Question

Use the definition of the Laplace transform to find F(s)=L{f(t)}(s) if f(t)={t1​ if 01.​

Answer 1

To find the Laplace transform of the given function, we need to split it into two cases and evaluate the integrals for each case.

Step-by-step explanation:

The Laplace transform of the function f(t) = {t^2/2 if 0<=t<1; 1 if t>=1;} can be found using the definition of the Laplace transform. The Laplace transform of f(t) is denoted as F(s) = L{f(t)}(s), where s is the complex variable.

To find F(s), we can split the function f(t) into two parts: one for t between 0 and 1, and another for t greater than or equal to 1. For 0 <= t < 1, the Laplace transform is given by:

`F(s) = ∫[0 to 1] (t^2/2)e^(-st) dt`

For t >= 1, the Laplace transform is given by:

`F(s) = ∫[1 to ∞] e^(-st) dt`

By evaluating these integrals, we can find the Laplace transform of f(t).