Derivative of Laplace Transform/Examples/Example 1
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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 |
$\blacksquare$
Sources
- 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Evaluation of Integrals: $45 \ \text{(a)}$