# Laplace Transforms

**Author**: John J Weber III, PhD
**Corresponding Textbook Sections**:

**Section 7.2** – Definition of the Laplace Transform

## Expected Educational Results

**Objective 19–1**: I understand the Laplace transform as a map of a function onto a specific class of functions.**Objective 19–2**: I can use the definition of Laplace transform to find the Laplace Transform of a given any function.

## Laplace Transforms

Definition: Laplace Transforms

Let be a function on . The **Laplace Transform** of is the function defined as

**Note**: You will need to know how to evaluate an improper integral; *l'Hopital's Rule* may be required.

**Example 01**: Find the Laplace transform of .

**Solution**:

By the definition of the Laplace Transform

Use the definition of improper integral:

Integrate by parts:

Rewrite rational expressions:

Use FTC-II:

Use Limit Laws:

Use l’Hopital’s Rule for the first two limits:

Evaluate limits:

So, .

**NOTE**: The more that you can recognize patterns of Laplace transforms, the more helpful Laplace transforms will be.

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**Last Modified**: Sunday, 8 November 2020 4:12 EDT