Laplace Transform Calculator (With Inverse & Piecewise Support)

A Laplace Transform Calculator is an online math tool used to convert time-domain functions into the frequency domain. It is widely used in engineering, physics, control systems, and differential equations.

This calculator helps you:

• Find Laplace transforms
• Solve inverse Laplace transforms
• Work with piecewise functions
• Evaluate integrals using Laplace methods
• Understand step-by-step solutions

What Is a Laplace Transform?

The Laplace transform converts a function of time $f(t)$f(t) into a function of $s$s:$$\mathcal{L}\{f(t)\} = \int_0^\infty e^{-st} f(t)\, dt$$L{f(t)}=∫0∞​e−stf(t)dt

This makes complex differential equations easier to solve.

Why Use a Laplace Transform Calculator?

✔ Saves time
✔ Reduces errors
✔ Shows steps
✔ Handles piecewise functions
✔ Supports inverse transforms
✔ Mobile friendly

How to Calculate Laplace Transform

1. Enter the function $f(t)$f(t)
2. Select transform type
3. Click Calculate
4. View result with steps

How to Calculate Inverse Laplace Transform

The inverse Laplace transform converts $F(s)$F(s) back into $f(t)$f(t):$$\mathcal{L}^{-1}\{F(s)\} = f(t)$$L−1{F(s)}=f(t)

Example:$$F(s) = \frac{1}{s-2} \Rightarrow f(t) = e^{2t}$$F(s)=s−21​⇒f(t)=e2t

Piecewise Laplace Transform Calculator

Piecewise functions are written using the unit step function:$$f(t) = \begin{cases} 0, & t < 2 \\ t, & t \ge 2 \end{cases}$$f(t)={0,t,​t<2t≥2​

Laplace transforms help solve these easily using:$$\mathcal{L}\{u(t-a)f(t-a)\}$$L{u(t−a)f(t−a)}

Laplace Transform of Common Functions

Function $f(t)$f(t)	Laplace Transform
1	$\frac{1}{s}$s1​
t	$\frac{1}{s^2}$s21​
$e^{at}$eat	$\frac{1}{s-a}$s−a1​
sin(at)	$\frac{a}{s^2 + a^2}$s2+a2a​
cos(at)	$\frac{s}{s^2 + a^2}$s2+a2s​

How to Calculate an Integral by Laplace Transform

Laplace transforms can solve integrals such as:$$\int_0^\infty e^{-st} \sin(at)\, dt = \frac{a}{s^2 + a^2}$$∫0∞​e−stsin(at)dt=s2+a2a​

This is useful in signal processing and system analysis.

Who Should Use This Calculator?

• Engineering students
• Physics learners
• Control system designers
• Math teachers
• Researchers

This Laplace Transform Calculator supports:

✔ Standard Laplace
✔ Inverse Laplace
✔ Piecewise functions
✔ Integral solutions
✔ Step-by-step answers

It is fast, accurate, and mobile-friendly.