Question

if f(t)=1 then L[f(t)is

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Answer 1

$\displaystyle \sf{ If \: f(t)=1 \: \: then \: \: L[f(t)] \: \: is \: \: \frac{1}{s} }$

Given :

f(t) = 1

To find :

L[f(t)]

Concept :

The LAPLACE TRANSFORMS of f(t), denoted by L{f(t)} and defined as :

$\displaystyle \sf{ L\{f(t)\} =\displaystyle \sf \int\limits_{0}^{\infty} e^{-st} \: f(t)\, dt}$

Solution :

Step 1 of 2 :

Write down the given function

Here the given function is f(t) = 1

Step 2 of 2 :

The Laplace transform of the function

The required Laplace transform

$\displaystyle \sf{ = L\{f(t)\} }$

$=\displaystyle \sf \int\limits_{0}^{\infty} e^{-st} \: f(t)\, dt$

$=\displaystyle \sf \int\limits_{0}^{\infty} e^{-st} .\: 1\, dt$

$=\displaystyle \sf \int\limits_{0}^{\infty} e^{-st} \, dt$

$=\displaystyle \sf \frac{ {e}^{ - st} }{ - s} \bigg| _{0}^{\infty}$

$=\displaystyle \sf 0 - \frac{ 1}{ - s}$

$=\displaystyle \sf \frac{ 1}{ s}$

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