Find a Laplace transform of $\mathcal{L}\{t^nf(t)\}$ with full solution

Question

I need to find a full solution to this transform: $\mathcal{L}\{t^nf(t)\}$.

I know the result, but don't know how to solve it.

Answer 1

Hint.

$$ \mathcal{L}(t f(t)) = -\frac{d}{ds}\int_0^{\infty}e^{-st}f(t) dt = -\int_0^{\infty}(-t)e^{-st}f(t)dt = \int_0^{\infty} e^{-st}(t f(t)) dt $$

then

$$ \mathcal{L}(t^n f(t)) = (-1)^n\frac{d^n}{ds^n}F(s) $$