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

$F\left(s\right)={\int}_{0}^{\mathrm{\infty}}{e}^{-st}f\left(t\right)dt$

where we assume s is a positive real number. For example, to find the Laplace transform of $f\left(t\right)={e}^{-t}$, the following improper integral is evaluated using integration by parts:

$F\left(s\right)={\int}_{0}^{\mathrm{\infty}}{e}^{-st}{e}^{-t}dt={\int}_{0}^{\mathrm{\infty}}{e}^{-(s+1)t}dt=\frac{1}{s+1}$

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

$f\left(t\right)=\mathrm{cos}at\to F\left(s\right)=\frac{s}{{s}^{2}+{a}^{2}}$