# Laplace Transform of t^2 by Cosine a t

## Theorem

Let $\sin$ denote the real sine function.

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

Then:

$\laptrans {t^2 \cos a t} = \dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3}$

## Proof

 $\ds \laptrans {t^2 \cos a t}$ $=$ $\ds -\map {\dfrac {\d^2} {\d s^2} } {\laptrans {\cos a t} }$ Higher Order Derivatives of Laplace Transform $\ds$ $=$ $\ds -\map {\dfrac {\d^2} {\d s^2} } {\dfrac a {s^2 + a^2} }$ Laplace Transform of Sine $\ds$ $=$ $\ds \dfrac {2 s^3 - 6 a^2 s} {\paren {s^2 + a^2}^3}$ Quotient Rule for Derivatives

$\blacksquare$