Derivative of Laplace Transform/Examples/Example 1

Example of Use of Derivative of Laplace Transform

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

$\ds \int_0^\infty t e^{-s t} \cos t \rd t$

$=$

$\ds \laptrans {t \cos t}$

Definition of Laplace Transform

$\ds $

$=$

$\ds -\frac \d {\d s} \laptrans {\cos t}$

$\ds $

$=$

$\ds -\map {\frac \d {\d s} } {\dfrac s {s^2 + 1} }$

$\ds $

$=$

$\ds \dfrac {s^2 - 1} {\paren {s^2 + 1}^2}$

$\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

$\blacksquare$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Evaluation of Integrals: $45 \ \text{(a)}$