Question

If f(t) is a function defined for all t ≥ 0, its Laplace transform F(s) is defined as
1. \(\mathop \smallint \nolimits_0^\infty {e^{st}}f\left( t \right)dt\)
2. \(\mathop \smallint \nolimits_0^\infty {e^{ - st}}f\left( t \right)dt\)
3. \(\mathop \smallint \nolimits_0^\infty {e^{ist}}f\left( t \right)dt\)
4. \(\mathop \smallint \nolimits_0^\infty {e^{ - ist}}f\left( t \right)dt\)

Answer 1

Correct Answer - Option 2 : \(\mathop \smallint \nolimits_0^\infty {e^{ - st}}f\left( t \right)dt\)

Concept:

This is a definition of Laplace transform

\(L\left\{ {f\left( t \right)} \right\} = \;\mathop \smallint \nolimits_0^\infty {e^{ - st}}f\left( t \right)dt = f\left( s \right)\)

Some important Laplace transforms are:

\(L\left( {{t^n}} \right) = \frac{{n!}}{{{s^{n + 1}}}}\)

\(L\left( {{t^n}} \right) = \frac{{n!}}{{{s^{n + 1}}}}\)

\(L\left( {{t^n}{e^{at}}} \right) = \frac{{n!}}{{{{\left( {s - a} \right)}^{n + 1}}}}\)