Laplace transforms A powerful tool in solving problems in engineering and physics is the Laplace transform. Given a function f(t), the Laplace transform is a new function F(s) defined by

where we assume then s is a positive real number. For example, to find the Laplace transform oj f(t)= e−t, the following improper integral is evaluated using integration by parts:

Verify the following Laplace transforms, where a is a real number.

Step 1:

We have to verify the Laplace transform of cos at=

Step 2:

By the definition of Laplace transform “Given a function f(t), the Laplace transform is a new function F(s) defined by