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If \(f(t)\) is a function defined for all \(t\ge 0\), its Laplace transform \(F(s)\) is defined as

1. \(\int_0^\infty e^{st}f(t)\,dt\)
2. \(\int_0^\infty e^{-st}f(t)\,dt\)
3. \(\int_0^\infty e^{ist}f(t)\,dt\)
4. \(\int_0^\infty e^{-ist}f(t)\,dt\)

Correct answer: \(\int_0^\infty e^{-st}f(t)\,dt\)

Solution

The Laplace transform of a function \(f(t)\) is defined by \(F(s)=\int_0^\infty e^{-st}f(t)\,dt\). The negative sign in the exponent is the standard convention for the unilateral Laplace transform.

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