Transformada de Laplace de t^n: demostración paso a paso con integración por partes

Published on October 26, 2026

Calculus

Se demuestra que la transformada de Laplace de t elevado a n, para n entero positivo, es n!/s^(n+1), usando integración por partes y una relación de recurrencia.

Question

Halla L{t^(n)} , n es un entero positivo. resolver paso a paso

Answer

Solución

Paso 1: Definición de la transformada.

Paso 2: Integración por partes. Tomamos

, de modo que

El término de frontera se anula: en

por

y en

porque

(para

). Por tanto

Paso 3: Caso base.

Paso 4: Aplicar la recurrencia repetidamente.

Como

, se obtiene

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