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/(s^{2}+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 s^{2}+1 = z.

∴ 2s ds=dz

t | z |

s | s^{2}+1 |

∞ | ∞ |

**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.

Find the Laplace transform of cos(t)/t.Summary:L{cos(t)/t} does NOT exist. |

**Also Read:**

Laplace transform of t: | 1/s^{2} |

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

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

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

Laplace transform of e^{-t}: | 1/(s+1) |

Laplace transform of 1: | 1/s |

## FAQs

**Q1: What is the Laplace transform of cos(at)/t?**

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

**Q2: Find the Laplace transform of cos at.**

Answer: The Laplace transform of cos(at) is a/(s^{2}+a^{2}).