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## Laplace transform calculator with steps

The Laplace transform calculator is used to convert the real variable function to a complex-valued function. This Laplace calculator provides the step-by-step solution of the given function.

By using our Laplace integral calculator, you can also get the differentiation and integration of the complex-valued function.

## How does the Laplace transformation calculator work?

Follow the below steps to transform a real-valued function.

•    Enter the function into the input box.

•    Use the keypad icon for entering the mathematics symbols.

•    Press the calculate button to get the result.

•    Click the show more button to view the solution with steps.

•    To enter another input hit the reset button.

## What is Laplace transformation?

Laplace transform is a method to convert the given function into some other function of s. It is an improper integral from zero to infinity of e to the minus st times f of t with respect to t. The notation of Laplace transform is an L-like symbol used to transform one function into another.

$L\left\{f\left(t\right)\right\}=F\left(s\right)$

Laplace transform converts the given real-valued function into a complex-valued function by integrating the function.

### The formula for Laplace Transform

The formula used for the transformation of the function is given below.

$F\left(s\right)=L\left\{f\left(t\right)\right\}=\int _0^{\infty \:}e^{-st}f\left(t\right)dt$

• f(t) is the function of t
• s is the frequency parameter of complex number.

## How to find the Laplace transform of a function?

Following are some examples solved by our Laplace solver.

Example 1

Find the Laplace transform of $f\left(t\right)=e^t+sin\left(t\right)$.

Solution

Step 1: Write the given functions equal to f(t) & g(t).

$f\left(t\right)=e^t$

$g\left(t\right)=sin\left(t\right)$

Step 2: Now Apply the Laplace notation.

$L\left\{f\left(t\right)+g\left(t\right)\right\}=L\left\{e^t+sin\left(t\right)\right\}$

Step 3: Now apply the linearity property of Laplace.

$L\left\{f\left(t\right)+g\left(t\right)\right\}=L\left\{e^t\right\}+L\left\{sin\left(t\right)\right\}$

Step 4: Now transform the functions by using the Laplace table.

$L\left\{e^t\right\}=\frac{1}{s-1}$

$L\left\{sin\left(t\right)\right\}=\frac{1}{s^2+1}$

Step 5: Put the values.

$L\left\{e^{t\:}+sin\left(t\right)\right\}=\frac{1}{s-1}+\frac{1}{s^2+1}$

Example 2

Find the Laplace transform of $e^{-2t}\sin ^2\left(t\right)$.

Solution

Step 1: Apply the notation of Laplace .

$L\left\{e^{-2t}\:sin\:^2\left(t\right)\right\}$

Step 2: Use a sine identity and put it into the given function.

We know that:

$\sin ^2\left(x\right)=\frac{1}{2}-\cos \left(2x\right)\frac{1}{2}$

So,

$L\left\{e^{-2t}\left(\frac{1}{2}-\cos \:\:\left(2t\right)\frac{1}{2}\right)\right\}$

Step 3: Now apply the linearity property of Laplace.

$L\left\{a\cdot \:\:f\left(t\right)+b\cdot \:\:g\left(t\right)\right\}=a\cdot \:L\left\{f\left(t\right)\right\}+b\cdot \:L\left\{g\left(t\right)\right\}$

$L\left\{e^{-2t}\left(\frac{1}{2}\cos \:\left(2t\right)\frac{1}{2}\right)\right\}=\frac{1}{2}L\left\{e^{-2t}\right\}-\frac{1}{2}L\left\{e^{-2t}\cos \:\:\left(2t\right)\right\}$

Step 4: Use the Laplace table to get the result.

$L\left\{e^{-2t}\right\}=\frac{1}{s+2}$

$L\left\{e^{-2t}cos\left(2t\right)\right\}=\frac{s+2}{\left(s+2\right)^2+4}$

Step 5: Put the values.

$L\left\{e^{-2t}\left(\frac{1}{2}-\cos \:\:\left(2t\right)\frac{1}{2}\right)\right\}=\frac{1}{2}\cdot \:\frac{1}{s+2}-\frac{1}{2}\cdot \:\frac{s+2}{\left(s+2\right)^2+4}$

$L\left\{e^{-2t}\left(\frac{1}{2}-\cos \:\:\left(2t\right)\frac{1}{2}\right)\right\}=\frac{2}{\left(s+2\right)\left(s^2+4s+8\right)}$

## Table of Laplace Transform

This Laplace Transform calculator with steps follows the below table to transform the functions.

 $f\left(t\right)$ $F\left(s\right)=L\left\{f\left(t\right)\right\}$ 1 $\frac{1}{s}$ $e^{at}$ $\frac{1}{s-a}$ $t^n,n=1,2,3,...$ $\frac{n}{s^{n+1}}$ $t^p, p>-1$ $\frac{Γ\left(p+1\right)}{s^{p+1}}$ $\sqrt{t}$ $\frac{\sqrt{\pi \:}}{2s^{\frac{3}{2}}}$ $t^{n-\frac{1}{2}},n=1,2,3,...\$ $\frac{1⋅3⋅5⋯\left(2n−1\right)\sqrt{\pi }}{2^ns^{n+\frac{1}{2}}}$ $sin\left(at\right)$ $\frac{a}{s^2+a^2}$ $cos\left(at\right)$ $\frac{s}{s^2+a^2}$ $tsin\left(at\right)$ $\frac{2as}{\left(s^2+a^2\right)^2}$ $tcos\left(at\right)$ $\frac{s^2-a^2}{\left(s^2+a^2\right)^2}$ $sin\left(at\right)−atcos\left(at\right)$ $\frac{2a^3}{\left(s^2+a^2\right)^2}$ $sin\left(at\right)+atcos\left(at\right)$ $\frac{2as^2}{\left(s^2+a^2\right)^2}$ $cos\left(at\right)−atsin\left(at\right)$ $\frac{s\left(s^2-a^2\right)}{\left(s^2+a^2\right)^2}$ $cos\left(at\right)+atsin\left(at\right)$ $\frac{s\left(s^2+)3a^2\right)}{\left(s^2+a^2\right)^2}$ $sin\left(at+b\right)$ $\frac{ssin\left(b\right)+acos\left(b\right)}{s^2+a^2}$ $cos\left(at+b\right)$ $\frac{scos\left(b\right)−asin\left(b\right)}{s^2+a^2}$ $sinh\left(at\right)$ $\frac{a}{s^2-a^2}$ $cosh\left(at\right)$ $\frac{s}{s^2-a^2}$ $e^{at}sin\left(bt\right)$ $\frac{b}{\left(s-a\right)^2+b^2}$ $e^{at}cos\left(bt\right)$ $\frac{s-a}{\left(s-a\right)^2+b^2}$ $e^{at}sinh\left(bt\right)$ $\frac{b}{\left(s-a\right)^2-b^2}$ $e^{at}cosh\left(bt\right)$ $\frac{s-a}{\left(s-a\right)^2-b^2}$ $t^ne^{at},\:n=1,2,3,...$ $\frac{n!}{\left(s-a\right)^{n+1}}$ $f\left(ct\right)$ $\frac{1}{s}F\left(\frac{s}{c}\right)$ $uc\left(t\right)=u\left(t−c\right)$ $e^{-cs}$ /s $δ\left(t−c\right)$ $e^{-cs}$ $u_c\left(t\right)f\left(t−c\right)$ $e^{-cs}F\left(s\right)$ $u_c\left(t\right)g\left(t\right)$ $e^{-cs}L\left\{g\left(t+c\right)\right\}$ $e^{ct}f\left(t\right)$ $F\left(s−c\right)$ $t^nf\left(t\right),n=1,2,3,...\$ $\left(-1\right)^nF^{\left(n\right)}\left(s\right)$ $\frac{1}{t}f\left(t\right)$ $\int _s^{\infty }F\left(u\right)du$ $\int _0^tf\left(v\right)dv$ F(s)/s $\int _0^tf\left(t−τ\right)g\left(τ\right)dτ$ $F\left(s\right)G\left(s\right)$ $f\left(t+T\right)=f\left(t\right)$ $\frac{\int _0^Te^{-st}f\left(t\right)dt\:}{1-e^{-sT}}$ $f'\left(t\right)$ $sF\left(s\right)−f\left(0\right)$ $f''\left(t\right)$ $s^2F\left(s\right)-s f\left(0\right)−f′\left(0\right)$

### References

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