Chapter 8.9, Problem 105E

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

$F(s)={\displaystyle {\int }_0^{\infty }{e}^{-st}f(t)dt},$

where we assume that s is a positive real number. For example, to find the Laplace transform of f(t) = e^−t, the following improper integral is evaluated:

$F(s)={\displaystyle {\int }_0^{\infty }{e}^{-st}{e}^{-t}dt={\displaystyle {\int }_0^{\infty }{e}^{-(s+1)t}dt=}\frac{1}{s+1}.}$

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

93. $f(t)=\sin at\to F(s)=\frac{a}{s^2+a^2}$