Inverse Laplace Transform Calculator

This Inverse Laplace Transform Calculator converts a function of s back into a function of t. Enter F(s) such as 1/(s-3) or 2/(s^2+4), and it returns f(t), using partial fractions when needed.

Step-by-step method

1. Write the transform F(s).
2. Apply partial fractions or a standard transform pair.
3. Write the inverse transform f(t).

Formula:

Example 1:

Step 1 - Write the transform F(s).

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

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

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

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

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

Final answer:

Example 2:

Step 1 - Write the transform F(s).

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

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

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

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

In this problem: The inverse Laplace transform is \(f(t) = \sin{\left(2 t \right)}\).

Final answer:

Example 3:

Step 1 - Write the transform F(s).

In this problem: We invert \(F(s) = \frac{1}{s^{2} + 3 s + 2}\).

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

In this problem: Decompose into partial fractions: \(F(s) = - \frac{1}{s + 2} + \frac{1}{s + 1}\).

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

In this problem: The inverse Laplace transform is \(f(t) = \left(e^{t} - 1\right) e^{- 2 t}\).

Final answer:

See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

1. Write the transform F(s).
2. Apply partial fractions or a standard transform pair.
3. Write the inverse transform f(t).

Step 1

Step 1 - Write the transform F(s).

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

\(\)

See Example 2 Hide Example 2

Problem

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

Approach

Step-by-step method

1. Write the transform F(s).
2. Apply partial fractions or a standard transform pair.
3. Write the inverse transform f(t).

Step 1

Step 1 - Write the transform F(s).

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

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

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 \(f(t) = \sin{\left(2 t \right)}\).

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

Final Answer

\(\)

See Example 3 Hide Example 3

Problem

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

Approach

Step-by-step method

1. Write the transform F(s).
2. Apply partial fractions or a standard transform pair.
3. Write the inverse transform f(t).

Step 1

Step 1 - Write the transform F(s).

In this problem: We invert \(F(s) = \frac{1}{s^{2} + 3 s + 2}\).

\(F(s) = \frac{1}{s^{2} + 3 s + 2}\)

Step 2

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

In this problem: Decompose into partial fractions: \(F(s) = - \frac{1}{s + 2} + \frac{1}{s + 1}\).

\(F(s) = - \frac{1}{s + 2} + \frac{1}{s + 1}\)

Step 3

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

In this problem: The inverse Laplace transform is \(f(t) = \left(e^{t} - 1\right) e^{- 2 t}\).

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

Final Answer

\(\)