# Derivative of Laplace Transform/Examples/Example 1

## Example of Use of Derivative of Laplace Transform

$\ds \int_0^\infty t e^{-2 t} \cos t \rd t = \dfrac 3 {25}$

## Proof

 $\ds \int_0^\infty t e^{-s t} \cos t \rd t$ $=$ $\ds \laptrans {t \cos t}$ Definition of Laplace Transform $\ds$ $=$ $\ds -\frac \d {\d s} \laptrans {\cos t}$ Derivative of Laplace Transform $\ds$ $=$ $\ds -\map {\frac \d {\d s} } {\dfrac s {s^2 + 1} }$ Laplace Transform of Cosine $\ds$ $=$ $\ds \dfrac {s^2 - 1} {\paren {s^2 + 1}^2}$ Quotient Rule for Derivatives $\ds \leadsto \ \$ $\ds \int_0^\infty t e^{-2 t} \cos t \rd t$ $=$ $\ds \dfrac {2^2 - 1} {\paren {2^2 + 1}^2}$ substituting $t = 2$ $\ds$ $=$ $\ds \dfrac 3 {25}$ evaluating

$\blacksquare$