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Prove: If L[F(t)] = f(s), then

Proof. Let

Then

1) G'(t) = F(t)

and G(0) = 0. Taking the Laplace transform of both sides gives

2) L[G'(t)] = L[F(t)] = f(s)

Also, from the formula for the transform of a derivative we have

3) L[G'(t)] = sL[G(t)] - G(0) = sL[G(t)]

From 2) and 3) we get

4) sL[G(t)] = f(s)

Thus

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