Where did I go wrong? [Laplace Transform]

[mathwurkz]

L is Laplace Transform operator.
Question is:

Let f(t) = t^2. Derive L[f] from L[1]

So I know f(1) =1 and L[1] = \frac{1}{s}

Carrying out the Transform...

L[f] = \int_{0}^{\infty} e^{-st}t^2 dt

Integration by parts u = t^2, dv = e^{-st} dt

I do the integration, the uv terms go to zero. after I clean up I get

- \int_{0}^{\infty} \frac{e^{-st}}{-s} 2t\ dt

Doing IBP again. u = t,\ dv = e^{-st} dt Rinse and repeat.

- \frac{2}{s} \int_{0}^{\infty} \frac{e^{-st}}{-s} (1) dt \\<br /> = - \frac{2}{s} \left( \frac{e^{-st}}{s^2}\right)_{0}^{\infty} = - \frac{2}{s^3}

Ok. So is that right? I don't think the negative is right at the end. But what the bigger thing that worries me is how do I use L[1] in this? Or is that a different approach?

 

[lurflurf]

It seems something like this is wanted
L[t^a \ f(t)]=\left(-\frac{d}{ds}\right)^aL[f]
thus
L[t^2]=L[t^2 \ 1]=\left(-\frac{d}{ds}\right)^2L[1]=\frac{d^2}{{ds}^2} \ \frac{1}{s}=\frac{2}{s^3}