Laplace Transform Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Pro

This Laplace Transform Calculator turns a function of time into a function of s, and shows the working line by line. The transform trades derivatives for multiplication, which is what makes a differential equation in t into an ordinary algebraic equation in s. Standard pairs cover the common functions, so the integral rarely has to be worked out by hand. Following the same pattern every time makes each transform easy to check against a table.

Step-by-step method

  1. Set up the problem.
  2. Apply the Laplace transform definition (or a standard transform pair).
  3. Write the transform F(s).

Formula:

\(\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\)
See Example 1 Hide Example 1

Problem

\(\mathcal{L}\{t^{3}\}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Apply the Laplace transform definition (or a standard transform pair).
  3. Write the transform F(s).

Step 1

Step 1 - Set up the problem.

In this problem: We transform \(f(t) = t^{3}\).

\(f(t) = t^{3}\)

Step 2

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

\(\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\)

Step 3

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{6}{s^{4}}\).

\(\mathcal{L}\{t^{3}\} = \frac{6}{s^{4}}\)

Final Answer

\(\mathcal{L}\{t^{3}\} = \frac{6}{s^{4}}\)
See Example 2 Hide Example 2

Problem

\(\mathcal{L}\{\cos{\left(3 t \right)}\}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Apply the Laplace transform definition (or a standard transform pair).
  3. Write the transform F(s).

Step 1

Step 1 - Set up the problem.

In this problem: We transform \(f(t) = \cos{\left(3 t \right)}\).

\(f(t) = \cos{\left(3 t \right)}\)

Step 2

Step 2 - Apply the Laplace transform definition (or a standard transform pair).

In this problem: Apply the Laplace transform to the function of t.

\(\mathcal{L}\{f(t)\} = F(s) = \int_{0}^{\infty} e^{-st} f(t) \, dt\)

Step 3

Step 3 - Write the transform F(s).

In this problem: The Laplace transform is \(F(s) = \frac{s}{s^{2} + 9}\).

\(\mathcal{L}\{\cos{\left(3 t \right)}\} = \frac{s}{s^{2} + 9}\)

Final Answer

\(\mathcal{L}\{\cos{\left(3 t \right)}\} = \frac{s}{s^{2} + 9}\)

Example Problems

Click one to load it into the box and solve it.