Finding Inverse Laplace transforms of a function

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Finding the inverse Laplace transform of the function

X(s) = s/{(s - 2)^1.5.(s^2 + 1)}

Using partial fraction expansion and inverse laplace transforms of known functions

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Question : Find the inverse Laplace transform, call it , of the function

Solution:

We will write the function in a different form noting that can be written
as thus

So now we let the term be denoted by and the term be denoted by

Now we look to expand into its constituent ...