Laplace Transform of Integral

Theorem

Let $f: \R \to \R$ or $\R \to \C$ be a function.

Let $\laptrans f = F$ denote the Laplace transform of $f$.

Then:

$\ds \laptrans {\int_0^t \map f u \rd u} = \dfrac {\map F s} s$

wherever $\laptrans f$ exists.

Let $\map g t = \ds \int_0^t \map f u \rd u$.

Then:

$\map {g'} t = \map f t$

and:

$\map g 0 = 0$

Thus:

$\ds \laptrans {\map {g'} t}$

$=$

$\ds s \laptrans {\map g t} - \map g 0$

$\ds $

$=$

$\ds s \laptrans {\map g t}$

$\ds $

$=$

$\ds \map F s$

as $\map F s = \laptrans {\map f t}$

$\ds \leadsto \ \ $

$\ds \laptrans {\map g t}$

$=$

$\ds \dfrac {\map F s} s$

$\ds \leadsto \ \ $

$\ds \laptrans {\int_0^t \map f u \rd u}$

$=$

$\ds \dfrac {\map F s} s$

$\blacksquare$

$\ds \laptrans {\int_0^1 \sin 2 u \rd u} = \dfrac 2 {s \paren {s^2 + 4} }$

1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Some Important Properties of Laplace Transforms: $6$. Laplace transform of integrals: Theorem $1 \text{-} 11$
1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Laplace Transform of Integrals: $17$
1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Appendix $\text A$: Table of General Properties of Laplace Transforms: $12.$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 32$: Table of General Properties of Laplace Transforms: $32.13$
2009: Murray R. Spiegel, Seymour Lipschutz and John Liu: Mathematical Handbook of Formulas and Tables (3rd ed.) ... (previous) ... (next): $\S 33$: Laplace Transforms: Table of General Properties of Laplace Transforms: $33.13$