 # 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

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## 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/(s5).

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! (s4+1)

Step 2: Now using the formula 1/s-n = en and 1/sn+1 = tn

L-1(s) = 9e-3t- e5/13/13 + 9/24 t4

Example 2:

Find the Laplace inverse transformation of the function s/(s2+7)+1/s2.

Solution:

Step 1: Taking the inverses separately we have

L-1(s/(s2+7)) = cos(71/2t)

L-1 (1/s2) = t

Step 2: Now combine the terms together, we have

L-1 (s/(s2+7)+1/s2)=cos(71/2t) + t

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