## Introduction to Inverse Laplace Transform Calculator with Steps

The inverse Laplace transform calculator is an advanced integral tool used to find the inverse of Laplace transform problems. It **evaluates complex variable** problems and converts the frequency domain into the time domain in less than a minute.

Our laplace inverse transform calculator is a beneficial tool for engineers, mathematics students, or researchers who want an easy way to solve inverse Laplace transform questions without any mistakes in fractions of a second.

The Inverse Laplace Transform Calculator provides users with a convenient and efficient way to perform this conversion. Users input the function F(s) in the Laplace domain, usually in the form of an algebraic expression, and the calculator computes the corresponding function f(t) in the time domain.

Additionally, if you would like to find z transform series of complex variables. You can be utilize our Z Transformation Calculator. Our calculator also evaluates a series that belongs to the real number or complex number into a discrete-time signal that can change into the frequency domain in real time.

## What is an Inverse Laplace Transform?

Inverse Laplace transform is defined as the **transformation of complex variable** functions F(s) to real variables f(t). Inverse Laplace is the reverse method of the Laplace transform. Inverse Laplace transformation changes the frequency domain problem into the time domain.

The inverse Laplace transform is an operation that takes a function of a complex variable (usually in the Laplace domain, which includes complex frequencies) and transforms it back into a function of time (or space) in the original domain.

Additionally, the inverse Laplace transform can be calculated using various techniques, such as partial fraction decomposition, contour integration, and tables of transforms, depending on the complexity of the function F(s) and the desired accuracy of the result. For a deeper dive into the Fourier transform and its application in analyzing functions in the frequency domain, you may utilize our fourier transform solver to gain insights into how functions can be transformed between the time and frequency domains.

## Formula of Inverse Laplace:

The **inverse Laplace formula** is opposite to the Laplace transform formula. The formula used by the inverse Laplace transform calculator with steps is as given,

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

In the integral form, the inverse Laplace transformation is given as,

$$ L^{-1} [F(s)] \;=\; \frac{1}{2 \pi i} \int_{a-i \infty}^{a+i \infty} F(s) e^{ts} ds $$

Whereas,

f(t) is the function of the time domain

F(s) is the function of the frequency domain

L-1 = shows the inverse Laplace transform function

This integral formula allows us to compute the inverse Laplace transform of a function F(s), yielding its corresponding function f(t) in the time domain. To further explore the relationship between time-domain functions and their frequency-domain representations, you can use our fourier series coefficients calculator to analyze periodic signals and decompose them into their constituent frequency components.

## Working Process of Laplace Inverse Transform Calculator

The laplace transform inverse calculator has a simple procedure to evaluate the inverse Laplace transform problems. It uses all the rules of the inverse Laplace transform to calculate the inverse Laplace transformation.

Our tool had an advanced server where these inverse Laplace formulas are built-in, so you can give any complex problem and it will **solve the given problem** immediately. When you give input of inverse Laplace questions in the calculator.

It analyze the given function and apply the formula of inverse Laplace after recognizing the function's nature. In a few seconds, you get the **solution of given inverse** problems. You can give various types of complex problems to convert into complex variables into real variables.

For further mathematical computations and to explore additional features, you can use our laplacian vector calculator. Our calculator provides not only inverse Laplace transform calculations but also Laplace transform calculations, providing a comprehensive tool for your mathematical needs.

Now let's understand an example and its solution to see the working process of inverse transform laplace calculator.

## Solved Example from Inverse Laplace

The inverse Laplace transform calculator will help you find the inverse Laplace transform in no time but it's crucial to understand each step so here is an example given,

### Example:

Find the following:

$$ \mathcal{L}^{-1} \biggr( \frac{3s+8}{s^2 + 2s + 5} \biggr) $$

**Solution:**

$$ e^{at} \leftrightarrow \frac{1}{s-a} \;and\; sin \mathscr{w} \leftrightarrow \frac{ \mathscr{w}}{s^2 + \mathscr{w}^2} $$

$$ \frac{3s+8}{s^2 + 2s + 5} \;=\; \frac{3s + 8}{(s+1)^2 +4} $$

$$ e^{-t} cos\; 2t \leftrightarrow \frac{s+1}{(s+1)^2 + 4} \;and\; e^{-t} sin\;2t\; \leftrightarrow \frac{2}{(s+1)^2 +4} $$

$$ \mathcal{L}^{-1} \biggr( \frac{3s+8}{(s+1)^2 +4} \biggr) \;=\; \mathcal{L}^{-1} \biggr( \frac{3s+3}{(s+1)^2 + 4} \biggr) + \mathcal{L}^{-1} \biggr( \frac{5}{(s+1)^2 + 4} \biggr) $$

$$ 3 \mathcal{L}^{-1} \biggr( \frac{s+1}{(s+1)^2 + 4} \biggr) + \frac{5}{2} \mathcal{L}^{-1} \biggr( \frac{2}{(s+1)^2 + 4} \biggr) $$

$$ e^{-t} \biggr( 3 cos\; 2t + \frac{5}{2} sin\;2t \biggr) $$

Thus it is the final solution of our function with specific limits. for further solve this example, you can use our laplace transformation calculator. Whether you need to calculate Laplace transforms or their inverses, our tool provides accurate results in seconds.

## How to Use the Inverse Laplace Transform Calculator?

The laplace inverse transform calculator is an **easy-to-use tool** as it has a user-friendly interface that provides its users with a comfortable experience that enables them to use it for the evaluation of Laplace transform problems. You need to follow some steps before calculation. These steps are:

- Enter the inverse Laplace transform function in its respective fields
- Click on the calculate button to get the solution of the inverse Laplace transform
- Press the recalculate button that brings you back to the home page where you can do more evaluation of inverse Laplace problems.

## Outcome from Laplace Transform Inverse Calculator

You **get the outcome** from the inverse Laplace transform calculator with steps immediately after you enter the input function in it. It may include as

For further mathematical computations, you can use our calculate integral online to simplify your integration tasks effortlessly. Whether you need to calculate definite or indefinite integrals, our tool provides accurate results in seconds.

- Result option provides you solution of inverse Laplace transform
- Possible steps section provides the solution of inverse Laplace in detail
- Plot section gives a graph after taking the values from the inverse Laplace solution

## Benefits of Inverse Transform Laplace Calculator

The Laplace transform inverse calculator gives you multiple **benefits** whenever you give the input value in it. These benefits are:

For an extensive collection of mathematical tools and calculators, including integral calculators, visit our all calculators section.

- Our tool keeps you away from doing lengthy calculations of Laplace transform questions
- The Laplace inverse transform calculator has a simple design so everyone can use it easily
- You do not need to spend a single penny for calculation in it.
- It is a trustworthy tool that always gives you precise solutions without any errors.
- It is a speedy calculator that provides a solution in a few seconds
- You can use it to solve complex variable functions of inverse Laplace
- The inverse Laplace transform calculator with steps provides the solution in the form of a graph for a better understanding.