Properties of Laplace Transform

Constant Multiple
If   $a$   is a constant and   $f(t)$   is a function of   $t$,   then
 

Example: $\mathcal{L} (4 \cos t) = 4 \, \mathcal{L} (\cos t)$
 

Linearity Property
If   $a$   and   $b$   are constants while   $f(t)$   and   $g(t)$   are functions of   $t$   whose Laplace transform exists, then
 

Example: $\mathcal{L} (3t^2 - 4t + 9) = 3 \, \mathcal{L} (t^2) - 4 \, \mathcal{L} (t) + 9 \, \mathcal{L} (1)$
 

First Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

Second Shifting Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   and   $g(t) = \begin{cases} f(t - a) & t \gt a \\ 0 & t \lt a \end{cases}$
 

then,

Change of Scale Property
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

Multiplication by Power of $t$
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

where   $n = 1, \, 2, \, 3, \, ...$
 

Division by $t$
If   $\mathcal{L} \left\{ f(t) \right\} = F(s)$,   then,
 

provided   $\displaystyle \lim_{t \rightarrow 0} \left[ \dfrac{f(t)}{t} \right]$   exists.
 

Transforms of Derivatives
The Laplace transform of the derivative   $f'(t)$   exists when   $s > a$,   and
 

In general, the Laplace transform of n^th derivative is