Laplace Transform of t by Sine a t/Proof 1

Theorem

Let $\sin$ denote the real sine function.

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

Then:

$\laptrans {t \sin a t} = \dfrac {2 a s} {\paren {s^2 + a^2}^2}$

Proof

\(\ds \laptrans {t \sin a t}\)

\(=\)

\(\ds -\map {\dfrac \d {\d s} } {\laptrans {\sin a t} }\)

Derivative of Laplace Transform

\(\ds \)

\(=\)

\(\ds -\map {\dfrac \d {\d s} } {\dfrac a {s^2 + a^2} }\)

Laplace Transform of Sine

\(\ds \)

\(=\)

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

Quotient Rule for Derivatives

$\blacksquare$

Sources

• 1965: Murray R. Spiegel: Theory and Problems of Laplace Transforms ... (previous) ... (next): Chapter $1$: The Laplace Transform: Solved Problems: Multiplication by Powers of $t$: $20 \ \text{(a)}$