# Laplace Transform of cos(t)/t

The Laplace transform of cos(t)/t does not exist as we get a divergent integral from the definition of the Laplace transform. In this article, we will discuss about L{cos(t)/t}.

## What is the Laplace Transform of cos(t)/t?

Answer: The Laplace transform of cos(t)/t does not exist.

Proof:

We know that the Laplace transform of a function f(t) divided by t, denoted by L{f(t)/t}, is given by the following division by t Laplace transform formula:

$L\{\frac{f(t)}{t} \} =\int_s^\infty F(s) ds$, where $L\{f(t)\}=F(s)$ …(I)

Step 1: Put f(t) = cos(t) in the above formula.

∴ F(s) = L{f(t)} = L{cos(t)} = s/(s2+1)

Step 2: Now, applying the formula (I), the Laplace transform of cos(t)/t is equal to

L{cos(t)/t} = $\int_s^\infty \dfrac{s}{s^2+1} ds$

Step 3: Put s2+1 = z.

∴ 2s ds=dz

Step 4: Then L{cos(t)/t} = $\dfrac{1}{2}\int_s^\infty \dfrac{2s}{s^2+1} ds$

= $\dfrac{1}{2}\int_{s^2+1}^\infty \dfrac{dz}{z} ds$

= $\dfrac{1}{2}$ $\Big[ \log(z)\Big]_{s^2+1}^\infty$ as the integration of dx/x is log x.

= $\dfrac{1}{2}$ $\Big[ \log \infty – \log(s^2+1)\Big]$

∞.

Thus, the Laplace transform of cos(t)/t is a divergent integral and so it does not exist. This is proved by the definition of Laplace transforms.