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# Inverse Laplace Transform Calculator

Enter the function into input box and click calculate button to calculate Laplace's inverse by using inverse Laplace transform calculator

## Inverse Laplace Transform Calculator

Inverse Laplace Transform calculator is used to evaluate the inverse Laplace transform of a given function. This Inverse Laplace calculator will give the result with steps.

## What is meant by inverse Laplace transform?

The inverse Laplace transform is the method to transform a given function into the function of time. Inverse Laplace transform is a mathematical technique for taking the inverse of a Laplace transform function. Inverse Laplace transform is also known as inverse Laplace or inverse Laplace transformation. It is denoted as L^{-1}(f(t)).

If F(s) is the Laplace transform of the function is f(t). then the inverse transformation is defined as an operator which maps f(t) to F(s). Symbolically it is written as

f(t) = L^{-1}(F(s)), F(s) = L(f(t))

**Formula:**

The inverse Laplace transformation in the integral form is given as:

$$L^{-1}\left[F\left(s\right)\right]=\frac{1}{2\pi i}\int _{\left(a-i∞\right)}^{\left(a+i∞\right)}F\left(s\right)\:e^{ts}\:ds$$

## Examples of Inverse Laplace Transform

We have to discuss some functions to find the Laplace inverse to understand the mathematical way how to calculate it.

**Example 1:**

Find the inverse Laplace transformation of the function 9/(s-3) + 1/(13s+5) +9/(s^{5}).

**Solution:**

**Step 1: **We have to first convert the given terms in the standard form to apply the formula:

L^{-1}(s) = 9/(s-(-3)) – 1/13(s-5/13) +9 (4!)/4! (s^{4+1})

**Step 2:** Now using the formula 1/s-n = e^{n} and 1/s^{n+1} = t^{n}

L^{-1}(s)** = **9e^{-3t}- e^{5/13}/13 + 9/24 t^{4}

**Example 2:**

Find the Laplace inverse transformation of the function s/(s^{2}+7)+1/s^{2}**.**

**Solution:**

**Step 1: **Taking the inverses separately we have

L^{-1}(s/(s^{2}+7))** = **cos(7^{1/2}t)

L^{-1 }(1/s^{2}) = t

**Step 2:** Now combine the terms together, we have

L^{-1 }(s/(s^{2}+7)+1/s^{2})=cos(7^{1/2}t) + t