The Laplace transform of cos2t/t is not defined. Here we will learn how to find the Laplace of cos2t/t. The Laplace transform formula of cos2t/t is given below.
L{cos2t/t} = undefined.
Table of Contents
Find the Laplace Transform of cos2t/t
Answer: The Laplace of cos2t/t does not exist.
Proof:
The following two formulas will be useful to find the Laplace of cos2t/t:
- L{cosat} = s/(s2+a2).
- $L\{\frac{f(t)}{t} \} =\int_s^\infty$ F(s) ds, where L{f(t)}=F(s)
Step 1: By formula (1), we have L{cos2t} = s/(s2+4).
Step 2: Put f(t)=cos2t in formula (2).
Thus, we get that
$L\{\frac{\cos 2t}{t} \} =\int_s^\infty \dfrac{s}{s^2+4} ds$ …(∗)
Step 3: Let s2+4 = u.
∴ 2s ds=du
| s | u |
| s | s2+4 |
| ∞ | ∞ |
Step 4: Form (∗), we have
$L\{\frac{\cos 2t}{t} \} =\int_s^\infty \dfrac{s}{s^2+4} ds$
= $\dfrac{1}{2}\int_{s^2+4}^\infty \dfrac{du}{u}$
= $\dfrac{1}{2}$ $\Big[ \log(u)\Big]_{s^2+4}^\infty$ as ∫dx/x = log x.
= $\dfrac{1}{2}$ $\Big[ \log \infty – \log(s^2+4)\Big]$
→ ∞, that is, it is a divergent integral.
So the Laplace transform of cos2t/t does not exist and it is proved by the definition of Laplace transforms.
ALSO READ:
| Laplace transform of 1 | 1/s |
| Laplace transform of t | 1/s2 |
| Laplace transform of sin t | 1/(s2+1) |
| Laplace transform of cos t | s/(s2+1) |
| Laplace transform of sin 2t/t | tan-1(2/s) |
FAQs
Answer: The Laplace transform of cos2t/t is undefined. That is, L{cos2t/t} does NOT exist.
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