Laplace transform of an integral of an unknown function of u

Question

What I've tried so far (for the first one):

$$\int^\infty_0e^{-sa} \int_0^t f(u)duda $$

$$= \int^\infty_0 \int_0^t e^{-sa} f(u)duda$$ $$= \int^t_0 \int_0^\infty e^{-sa} f(u)dadu$$

And now the inner integral looks like Laplace transform of $f(u)$ subject to the outer integral. I'm not sure if this is at all the right track, and if it is, I'm not sure how to proceed. Thanks in advance for any help.

Answer 1

Here is one way of evaluating the first transform:

let $g(t) = \int_0^t f(u) du$, then we have $\dot{g}(t) = f(t)$, with $g(0) = 0$, and taking the transform of both sides gives $s \hat{g}(s) - g(0) = \hat{f}(s)$, or, more simply, $\hat{g}(s) = {1 \over s} \hat{f}(s)$.