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/(s
^{2}+a^{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/(s^{2}+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 s^{2}+4 = u.

∴ 2s ds=du

s | u |

s | s^{2}+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/s^{2} |

Laplace transform of sin t | 1/(s^{2}+1) |

Laplace transform of cos t | s/(s^{2}+1) |

Laplace transform of sin 2t/t | tan^{-1}(2/s) |

## FAQs

**Q1: What is the Laplace transform of cos2t/t.**

Answer: The Laplace transform of cos2t/t is undefined. That is, L{cos2t/t} does NOT exist.