Can you integrate by parts with one integral inside another? From the definition of the Laplace transform:
$$\mathcal{L}[f(t)]\equiv \int_{t=0}^{\infty}f(t)e^{-st}\mathrm{d}t$$
where $s \in \mathbb{R^+}$. 
$$\mathcal{L}\left[\int_{u=0}^{t}f(u)\mathrm{d}u\right]=\int_{t=0}^{\infty}e^{-st}\mathrm{d}t\int_{u=0}^{t}f(u)\mathrm{d}u$$
$$=\int_{t=0}^{\infty}e^{-st}\mathrm{d}t\int_{u=0}^{t} f(u)\mathrm{d}u=\int_{u=0}^{t} f(u) \mathrm{d}u\int_{t=0}^{\infty}e^{-st}\mathrm{d}t$$
$$\left[-\cfrac{1}{s}e^{-st} \int_{u=0}^{t} f(u)\mathrm{d}u \right]_{\color{red}{0}}^{\color{red}{\infty}}+\int_{\color{red}{0}}^{\color{red}{\infty}}\cfrac{1}{s}e^{-st}f(\color{blue}{t})\mathrm{d}\color{blue}{t}$$
$$=\cfrac{1}{s}\mathcal{L}\left[\color{#180}{f}\right]$$
Assuming that on page 486 in this book it was done by parts I have $\mathbf{3}$ questions about the calculation above:
$\mathbf{\color{red}{1)}}$ For the $2$ sets of limits marked red above for which variable ($u$ or $t$) do they belong?
$\mathbf{\color{blue}{2)}}$ For the variables marked blue, shouldn't the $t$'s be $u$'s?
$\mathbf{\color{#180}{3)}}$ For the part marked green should this be $f(t)$ instead of $t$?
Please explain your answers.
Thank you.

1) $t$
2) No: The $f(t)$ there should be thought of as ${d\over dt}\int_0^t f(u)\,du$, in accordance with the integration by parts.
3) No, $\mathcal L[f]$ is correct.
