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.

Problem 78E

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 of , the following improper integral is evaluated using integration by parts:

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

Step 1</p>

In this problem we have to verify the given laplace transform.

That is we have to prove : where a is a real number.

The laplace transform of a function is defined as follows.

Given a function f(t), the Laplace transform is a new function F(s) defined by where we assume s is a positive real number.

Step 2</p>

Here we have

Thus … (1)

We shall integrate this by using integration by parts technique.

Integration by parts is a technique for performing indefinite integration , by expanding the differential of a product of functions and expressing the original integral in terms of a known integral . Thus we get

… (2)