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 79ELaplace 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.SolutionStep 1In 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.