Laplace Transform of Integral/Examples/Example 1

Examples of Use of Laplace Transform of Integral

Let $\laptrans f$ denote the Laplace transform of the real function $f$.

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

Proof

\(\ds \laptrans {\int_0^1 \sin 2 u \rd u}\)

\(=\)

\(\ds \dfrac 1 s \laptrans {\sin 2 t}\)

Laplace Transform of Integral

\(\ds \)

\(=\)

\(\ds \dfrac 2 {s \paren {s^2 + 2^2} }\)

Laplace Transform of Sine

\(\ds \)

\(=\)

\(\ds \dfrac 2 {s \paren {s^2 + 4} }\)

simplification

$\blacksquare$

Sources

• 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