and for , where   is continuous.

is the Laplace transform of and is the Laplace transform of .

Therefore from the Laplace transform definition .

Definition of improper integral type 1:.

.

Find the integral by using integration by parts.

Integration by parts: .

Let and

Find by integrating .

.

Differentiate on each side.

Substitute corresponding values in the by parts formula.

Consider , such that .

If , then for all values of .

.

Multiply on each side by .

Apply infinite limit on each side.

.

Find :

If , then .

.

Thus, .

Substitute in

for .

for .