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Find the inverse Laplace transform?

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Factor the denominator: s2 +3s - 4 = (s+4)(s-1). You can make sure that the function can be rewritten as

                                    F(s) = (2/5)[ 1/(s-1) - 1/(s+4)]               


                                 L-1 F(s) = (2/5) L-1 {1/(s-1)} - L-1{1/(s+4)}

Now, if you use the table for the inverse Laplace transforms you will come up exactly with the answer you need. 

If you want to go over integration, you will need to use Jordan's theorem and apply   res {F(s)},

where s is considered to be a complex variable, and F(s) has two singular points on the real axis:

                                                     s = -4   and s =1

Tell me if you need to show this approach to the problem and I'll get back to this, so you can find the inverse Laplace transform not refering to tables.