Laplace Transform: Order of transformation

1. Jan 29, 2010

libelec

1. The problem statement, all variables and given/known data
Find the Laplace transform for $$$t\left( {\int\limits_0^{t - a} {f(u)du} } \right)H(t - a)$$$ proving the properties used.

2. Relevant equations

If f(t) transforms into F(s):

a) $$${( - t)^{n}f(t) \to {F^{(n)}}(s)$$$

b) $$$\int\limits_0^t {f(u)du} \to \frac{{F(s)}}{s}$$$

c) $$$f(t - a)H(t - a) \to {e^{ - as}}F(s)$$$

3. The attempt at a solution

Here's my question: in what order should I use these properties? There's only one posibility for the translation, and that's that the transform is multiplied by e-as. But with the derivative of F(s) and the division for s there are two posibilities:

1) That the Laplace transform of $$$t\left( {\int\limits_0^{t - a} {f(u)du} } \right)H(t - a)$$$ be $$$\frac{{ - {e^{ - as}}{F^'}(s)}}{s}$$$.

2) That the Laplace transform be $$$- {e^{ - as}}{\left[ {\frac{{F(s)}}{s}} \right]^'}$$$.

What's the correct one and why? I couldn't find a reason to use either one but not the other.

EDIT: In property a), the lineal variable t should have an exponent without brackets, since it's not the n-th derivative but the n-th power.