The Laplace transform of (1-cost)/t is equal to 1/2 log[(s2+1)/s2]. Here we find Laplace of (1-cost)/t.
The Laplace transform formula of (1-cost)/t is given by
L{(1-cost)/t} = 1/2 log[(s2+1)/s2].
Table of Contents
Find the Laplace transform of (1-cost)/t
| Answer: 1/2 log[(1+s2)/s2] is the Laplace transform of (1-cost)/t. |
Solution:
Here we will use the following formula:
L$\Big[ \dfrac{f(t)}{t} \Big]$ = $\int_s^\infty F(s) ds$ …(I)
where F(s) = L{f(t)}, the Laplace transform of f(t).
To find the Laplace transform of (1-cost)/t, let us put f(t) = 1-cost in the above formula (I).
So F(s) = L{f(t)} = L{1-cost}
= L{1} – L{cost}
= $\dfrac{1}{s}$ – $\dfrac{s}{s^2+1}$ as the Laplace of cos(at) is s/(s2+a2).
Now, from the formula (I), we have that
L{(1-cost)/t} = $\int_s^\infty \Big[ \dfrac{1}{s} – \dfrac{s}{s^2+1} \Big] ds$
= $\Big[ \log s – \dfrac{1}{2} \log(1+s^2) \Big]_s^\infty$
= $\dfrac{1}{2}\Big[ 2\log s – \log(1+s^2) \Big]_s^\infty$
= $\dfrac{1}{2}\Big[ \log \dfrac{s^2}{1+s^2} \Big]_s^\infty$
= $\dfrac{1}{2}$ lims→∞ log $\dfrac{s^2}{1+s^2}$ – $\dfrac{1}{2} \log \dfrac{s^2}{1+s^2}$
= $\dfrac{1}{2}$ log 1 + $\dfrac{1}{2} \log \dfrac{1+s^2}{s^2}$
= $\dfrac{1}{2} \log \dfrac{1+s^2}{s^2}$ as the value of log 1 equals 0.
So the Laplace transform of (1-cost)/t is equal to 1/2 log[(1+s2)/s2]. and this is obtained by the division by t formula of Laplace transforms.
More Laplace Transforms:
Main Article: Laplace Transform: Definition, Table, Formulas, Properties
Laplace tansform of a constant
Find Laplace transform of cos2t
FAQs
Answer: The Laplace of (1-cost)/t is 1/2 log[(1+s2)/s2].
