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The unilateral Laplace transform of \(f(t)\) is \(\dfrac{1}{s^2+s+1}\). The unilateral Laplace transform of \(t f(t)\) is

1. \(-s/(s^2+s+1)^2\)
2. \(-(2s+1)/(s^2+s+1)^2\)
3. \(s/(s^2+s+1)^2\)
4. \((2s+1)/(s^2+s+1)^2\)

Correct answer: \((2s+1)/(s^2+s+1)^2\)

Solution

If \(F(s)=\mathcal{L}\{f(t)\}=\dfrac{1}{s^2+s+1}\), then \(\mathcal{L}\{t f(t)\}=-\dfrac{dF}{ds}\). Differentiating gives \(-\left[-\dfrac{2s+1}{(s^2+s+1)^2}\right]=\dfrac{2s+1}{(s^2+s+1)^2}\).

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