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Find the inverse Laplace transform?

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I found an alternative answer.

We can first factor (2s - 3)/(s2 - 4) into A/(s+2) + B(s - 2). By the solution methods in your book, you should get A = 7/4 and B = 1/4. We know that F(s) = A/(s - a) means that f(t) = e-at, so what you should get for the answer is f(t) = 7/4 * e-2t + 1/4 * e2t.