Laplace Transform of cos2t/t

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.

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:

1. L{cosat} = s/(s²+a²).
2. $L\{\frac{f(t)}{t} \} =\int_s^\infty$ 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.

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Laplace transform of cos t	s/(s2+1)
Laplace transform of sin 2t/t	tan-1(2/s)

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