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If \(F(s)\) is the Laplace transform of the function \(f(t)\), then the Laplace transform of \(\int_0^t f(\tau)\,d\tau\) is

1. \(\frac{1}{s}F(s)\)
2. \(\frac{1}{s}F(s)-f(0)\)
3. \(sF(s)-f(0)\)
4. \(\int F(s)\,ds\)

Correct answer: \(\frac{1}{s}F(s)\)

Solution

If \(g(t)=\int_0^t f(\tau)d\tau\), then \(g'(t)=f(t)\) and \(g(0)=0\). Taking Laplace transforms gives \(sG(s)-g(0)=F(s)\), so \(G(s)=F(s)/s\).

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