The inverse Laplace transform calculator is a digital tool that works backward to reveal the original mathematical expressions (real variables) from a transformed version called Laplace Transform. This tool aids in understanding the behavior of systems described in the Laplace domain.

## Concepts of Inverse Laplace Transform:

The inverse Laplace transform is a method in which we can revert back from the __frequency__ domain F(s) to the time domain f(t).

Inverse Laplace Transforms are, as its name suggests, the inverse operation of Laplace Transforms.

The Laplace Transform helps you change a function from a time domain f(t) to a frequency domain F(s). On the other hand, the inverse Laplace Transform reverses this transformation.

Inverse Laplace transform is also identified as inverse Laplace transformation or inverse Laplace and expressed as $$ L^{-1}f(t) $$.

## Inverse Laplace Transform Expressions:

Here's the formula to calculate the inverse Laplace transform,

$$ f(t) = \mathcal{L}^{-1}F(s) $$

In this formula:

$$ {f}{(t)} $$ represents the original function in the time domain.

$$ L^{-1} $$ denotes the inverse Laplace transform.

$$ {F}{(s)} $$ stands for the transformed function in the Laplace domain.

## Laplace Inverse Table:

Here's a table of common examples of inverse Laplace transforms,

Laplace Transform (F(s)) | Inverse Laplace Transform (f(t)) |
---|---|

1 | δ(t) (Dirac delta function) |

e^(at) | 1/(s-a) |

t^n (for n ≥ 0) | n!/(s^(n+1)) |

e^(at)sin(bt) | b/((s-a)^2 + b^2) |

e^(at)cos(bt) | (s-a)/((s-a)^2 + b^2) |

1/(s-a) | e^(at) |

1/(s^2) | t |

1/(s(s-a)) | 1 - e^(at) |

## Inverse Laplace Transform Calculations & Examples:

We are going to solve an example through which you can easily calculate the inverse Laplace transform.

### Example :

Let's find the inverse Laplace transform:

$$ F(s) = \frac{12}{s} - \frac{1}{s - 10} + 8(s - 25) $$

Upon examination of the terms, we notice that the denominator of the first term is simply a constant. The correct numerator for this term is "1". When using the inverse Laplace Transform Calculator with steps, we will only consider the factor 12 before performing the inverse transformation. Therefore, **a = 10** is the correct numerator, just as it needs to be.

The third term appears to be exponential, but this time, **a = 25**. Before performing the inverse transformation, we need to factor in the 8.

In more detailed form:

$$ F(s) = \frac{12}{s} - \frac{1}{s - 10} + 8(s - 25) f(t) = 12(1) - e^{10t} + 8e^{25t} = 12 - e^{10t} + 8e^{25t} $$

So, the inverse transform of $$ {F}{(s)} is {f}{(t)} = 12 - e^{10t} + 8e^{25t} $$. You can also try the inverse Laplace calculator for hassle-free and faster results.

## Steps to Use Inverse Transform Calculator:

If you don’t want to solve manually then you can use this free online inverse laplace transform calculator for this. All you need to do is follow simple steps which include:

**What to do:**

- Enter a complex function F(s) and see the equation preview in Laplace form.
- Press the Calculate button.

**What you get: **

- The inversed version of the given Laplace equation.

## FAQs:

### What are the properties of inverse Laplace?

Here are some properties that describe the inverse Laplace of a function.

- Linearity Property.
- Change of Scale Property.
- Shifting Property
- Second Shifting Property
- Property of Inverse Laplace Transform of Integrals.
- Property of Inverse Laplace Transform of Derivatives.
- Property of Multiplication by the Powers of s.

### What are the different types of Laplace transforms?

There are two types of Laplace transformation such as,

- One-sided Laplace transformation
- Two-sided Laplace transformation

## References:

**Libretexts.org:** The Inverse Laplace Transform and Definition of the Inverse Laplace Transform