The Laplace transform of t sint is 2s/(s^{2}+1)^{2}, that is, L{t sint}=2s/(s^{2}+1)^{2}. The function t sint is the product of t and the sine of t. In this article, we will find the Laplace transform of both t sint and t sin(at).

Table of Contents

## What is the Laplace Transform of t sin(t)?

**Answer:** The Laplace transform of t sin t is 2s/(s^{2}+1)^{2}.

*Proof:*

To find the Laplace transform of t sin(t), we will use the formula for the Laplace transform of a function f(t) multiplied by t, denoted by L{t f(t)}, which is given by the following formula: (multiplication by t Laplace transform formula)

$L\{t f(t)\} = – \dfrac{d}{ds}(F(s))$, where L{f(t)}=F(s) **…(∗)**

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

∴ F(s) = L{f(t)} = L{sin t} = 1/(s^{2}+1).

**Step 2:** So the Laplace transform of tsin(t) by **(∗)** is equal to

$L\{t\sin t\} = – \dfrac{d}{ds}\left(\dfrac{1}{s^2+1}\right)$

**Step 3:** By quotient rule of derivatives, we obtain that

$L\{t\sin t \}$ $= – \dfrac{(s^2+1)\frac{d}{ds}(1)-1 \frac{d}{ds}(s^2+1)}{(s^2+1)^2}$

$= – \dfrac{(s^2+1)\cdot 0- 1\cdot 2s}{(s^2+1)^2}$

$= – \dfrac{-2s}{(s^2+1)^2}$

$= \dfrac{2s}{(s^2+1)^2}$.

So the Laplace transform of tsin t is 2s/(s^{2}+1)^{2}.

Find the Laplace transform of t sint.Summary:L{t sin t} = 2s/(s ^{2}+1)^{2}. |

**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 t cost: | (s^{2}-1)/(s^{2}+1)^{2} |

Laplace transform of cos(t)/t: | Does Not Exist |

## What is the Laplace Transform of t sin(at)?

**Answer:** The Laplace transform of t sin at is 2as/(s^{2}+a^{2})^{2}.

*Proof:*

Put f(t) = t sin(at) in the above formula **(∗)**.

Note that L{sin at} = a/(s^{2}+a^{2}). So the Laplace transform of t sin(at) by the above formula **(∗)** is equal to

$L\{t\sin(at)\} = – \dfrac{d}{ds}\left(\dfrac{a}{s^2+a^2}\right)$

$= – \dfrac{(s^2+a^2)\frac{d}{ds}(a)-a \frac{d}{ds}(s^2+a^2)}{(s^2+a^2)^2}$

$= – \dfrac{(s^2+a^2)\cdot 0-a \cdot 2s}{(s^2+a^2)^2}$

$= – \dfrac{-2as}{(s^2+a^2)^2}$

$= \dfrac{2as}{(s^2+a^2)^2}$.

So the Laplace transform of tsin at is 2as/(s^{2}+a^{2})^{2}.

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

Laplace transform of 1 | 1/s |

Laplace transform of 1/t | Not Exist |

## FAQs

**Q1: t sint Laplace transform.**

Answer: The Laplace transform of the product tsint is 2s/(s^{2}+1)^{2}, that is, L{t sin t} = 2s/(s^{2}+1)^{2}.

**Q2: Find the Laplace transform formula of t sin(at).**

Answer: The Laplace transform of the product tsinat is 2as/(s^{2}+a^{2})^{2}, that is, L{t sin at} = 2as/(s^{2}+a^{2})^{2}.

This article is written by Dr. T. Mandal, Ph.D in Mathematics. On Mathstoon.com you will find Maths from very basic level to advanced level. Thanks for visiting.