Inverse Laplace Transform Notes

Published on: September 29, 2026

This Inverse Laplace Transform Calculator turns a function of s back into a function of time, and shows the working line by line. A transform that is not already a standard pair is split by partial fractions first, since each piece on its own usually is one. Each piece is then matched to the function it came from. Following the same pattern every time keeps the split and the matching from being run together.

Step-by-step method

  1. Set up the problem.
  2. Apply partial fractions or a standard transform pair.
  3. Write the inverse transform f(t).

Formula:

\(\mathcal{L}^{-1}\{F(s)\} = f(t)\)
See Example 1 Hide Example 1

Problem

\(\mathcal{L}^{-1}\{\frac{1}{s - 3}\}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Apply partial fractions or a standard transform pair.
  3. Write the inverse transform f(t).

Step 1

Step 1 - Set up the problem.

In this problem: We invert \(F(s) = \frac{1}{s - 3}\).

\(F(s) = \frac{1}{s - 3}\)

Step 2

Step 2 - Apply partial fractions or a standard transform pair.

In this problem: Match F(s) to a standard transform pair.

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

Step 3

Step 3 - Write the inverse transform f(t).

In this problem: The inverse Laplace transform is \(f(t) = e^{3 t}\).

\(\mathcal{L}^{-1}\{\frac{1}{s - 3}\} = e^{3 t}\)

Final Answer

\(\mathcal{L}^{-1}\{\frac{1}{s - 3}\} = e^{3 t}\)
See Example 2 Hide Example 2

Problem

\(\mathcal{L}^{-1}\{\frac{s}{s^{2} + 9}\}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Apply partial fractions or a standard transform pair.
  3. Write the inverse transform f(t).

Step 1

Step 1 - Set up the problem.

In this problem: We invert \(F(s) = \frac{s}{s^{2} + 9}\).

\(F(s) = \frac{s}{s^{2} + 9}\)

Step 2

Step 2 - Apply partial fractions or a standard transform pair.

In this problem: Match F(s) to a standard transform pair.

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

Step 3

Step 3 - Write the inverse transform f(t).

In this problem: The inverse Laplace transform is \(f(t) = \cos{\left(3 t \right)}\).

\(\mathcal{L}^{-1}\{\frac{s}{s^{2} + 9}\} = \cos{\left(3 t \right)}\)

Final Answer

\(\mathcal{L}^{-1}\{\frac{s}{s^{2} + 9}\} = \cos{\left(3 t \right)}\)
See Example 3 Hide Example 3

Problem

\(\mathcal{L}^{-1}\{\frac{2}{s + 4}\}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Apply partial fractions or a standard transform pair.
  3. Write the inverse transform f(t).

Step 1

Step 1 - Set up the problem.

In this problem: We invert \(F(s) = \frac{2}{s + 4}\).

\(F(s) = \frac{2}{s + 4}\)

Step 2

Step 2 - Apply partial fractions or a standard transform pair.

In this problem: Match F(s) to a standard transform pair.

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

Step 3

Step 3 - Write the inverse transform f(t).

In this problem: The inverse Laplace transform is \(f(t) = 2 e^{- 4 t}\).

\(\mathcal{L}^{-1}\{\frac{2}{s + 4}\} = 2 e^{- 4 t}\)

Final Answer

\(\mathcal{L}^{-1}\{\frac{2}{s + 4}\} = 2 e^{- 4 t}\)
See Example 4 Hide Example 4

Problem

\(\mathcal{L}^{-1}\{\frac{3}{s^{2} + 9}\}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Apply partial fractions or a standard transform pair.
  3. Write the inverse transform f(t).

Step 1

Step 1 - Set up the problem.

In this problem: We invert \(F(s) = \frac{3}{s^{2} + 9}\).

\(F(s) = \frac{3}{s^{2} + 9}\)

Step 2

Step 2 - Apply partial fractions or a standard transform pair.

In this problem: Match F(s) to a standard transform pair.

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

Step 3

Step 3 - Write the inverse transform f(t).

In this problem: The inverse Laplace transform is \(f(t) = \sin{\left(3 t \right)}\).

\(\mathcal{L}^{-1}\{\frac{3}{s^{2} + 9}\} = \sin{\left(3 t \right)}\)

Final Answer

\(\mathcal{L}^{-1}\{\frac{3}{s^{2} + 9}\} = \sin{\left(3 t \right)}\)