# 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

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.