first shifting

Problem 04 | Evaluation of Integrals

Problem 04
Evaluate   $\displaystyle \int_0^\infty \dfrac{e^{-t}\sin t}{t} ~ dt$.
 

Solution 04
From Problem 01 | Division by t:
$\mathcal{L} \left( \dfrac{\sin t}{t} \right) = \arctan \left( \dfrac{1}{s} \right)$
 

By first shifting property:

Problem 03 | Evaluation of Integrals

Problem 03
Find the value of   $\displaystyle \int_0^\infty te^{-3t} \sin t ~ dt$
 

Problem 04 | First Shifting Property of Laplace Transform

Problem 04
Find the Laplace transform of   $f(t) = e^t \sinh 2t$.
 

Solution 04

Problem 03 | First Shifting Property of Laplace Transform

Problem 03
Find the Laplace transform of   $f(t) = e^{-3t} \cos t$.
 

Solution 03

Problem 02 | First Shifting Property of Laplace Transform

Problem 02
Find the Laplace transform of   $f(t) = e^{-5t} \sin 3t$.
 

Solution 02

Problem 01 | First Shifting Property of Laplace Transform

Problem 01
Find the Laplace transform of   $f(t) = e^{2t}t^3$.
 

Solution 01

First Shifting Property | Laplace Transform

First Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   when   $s > a$   then,
 

$\mathcal{L} \left\{ e^{at} \, f(t) \right\} = F(s - a)$

 

In words, the substitution   $s - a$   for   $s$   in the transform corresponds to the multiplication of the original function by   $e^{at}$.
 

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