Evaluation of Integral using Laplace Transform/Examples/Example 1

Examples of Use of Evaluation of Integral using Laplace Transform

 * $\displaystyle \int_0^\infty \dfrac {e^{-t} - e^{-3 t} } t \rd t = \ln 3$

Proof
Let $\map f t = e^{-t} - e^{-3 t}$.

Then from Laplace Transform of Exponential:


 * $\laptrans {\map f t} = \dfrac 1 {s + 1} - \dfrac 1 {s + 3}$

Hence: