For f(t) defined for t0, its Laplace transform F(s) is .

$\int_0^\infty e^{ist}f(t)\,dt$

$\int_0^\infty e^{-ist}f(t)\,dt$

GATE 2016 ME (ME1) – Question 12

Engineering Mathematics · Differential Equations: Initial and boundary value problems, Laplace transforms · 1 mark · Multiple choice

For $f(t)$ defined for $t\ge0$, its Laplace transform F(s) is ______.

$\int_0^\infty e^{st}f(t)\,dt$
$\int_0^\infty e^{-st}f(t)\,dt$
$\int_0^\infty e^{ist}f(t)\,dt$
$\int_0^\infty e^{-ist}f(t)\,dt$

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

Explanation

For a function on t≥0, the Laplace transform is the integral from0 to infinity of exp(−st)f(t)dt, where it converges. The decaying exponential kernel distinguishes it from the positive-exponent and Fourier-type choices. Thus **B**.