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Here you will find the Laplace transform of functions as well as their inverse Laplace transform. The Laplace transform table, formula, properties, and applications to find the solution of an ODE will be discussed here.

**Main Article:** Laplace Transform: Definition, Table, Formulas, Properties

Laplace Transform of Derivatives

Laplace Transform of Integrals

Laplace of 1/t does NOT exist: Proof

For the Laplace transforms of your desired functions, just click on the function, and we will get your answer.

**Laplace transforms of Basic Functions: **

Functions f(t) | L{f(t)} |

1 | L{1} = 1/s |

t | L{t} = 1/s^{2} |

t^{3} | L{t^{3}} = 6/s^{4} |

t^{n} | L{t^{n}} = n!/s^{n+1} |

2^{t} | L{2^{t}} =1/(s-ln2), s>ln2 |

e^{at} | L{e^{at}} = 1/(s-a) |

sint | L{sint} = 1/(s^{2}+1) |

sinat | L{sin at} = a/(s^{2}+a^{2}) |

cost | L{cost} = s/(s^{2}+1) |

cosat | L{cos at} = s/(s^{2}+a^{2}) |

**More Laplace transforms:**

te^{t} | t sint | t cost |

e^{t}/t | sint/t | cost/t |

te^{at} | t sinat | t cosat |

sin2t/t | cos2t/t | cos^{2}t |

sin^{2}t | (1-cost)/t | (1-e^{t})/t |

sin2t sin3t | sint sin2t sin3t | t^{2}sin2t |

t sin2t | sin^{3}t | |

sinh(at) | cosh(at) |

Inverse Laplace Transforms

Table of Inverse Laplace Transformations

Inverse Laplace transform of 1