In that formula :
$$ u(t)=K_{\mathrm{P}} e(t)+K_{\mathrm{I}} \int_{0}^{t} e(\tau) \mathrm{d} \tau+K_{\mathrm{D}} \frac{\mathrm{d} e(t)}{\mathrm{d} t} $$
I know from the formulae collection we refer to that : $$ \mathcal{L} \left\{ \int_{0}^{t} e(\tau) \mathrm{d} \tau \right\} = \frac{E(s)}{s} $$
But I don't manage to find a proof. I watched a few videos on convolution theorem with two functions, but they are general, I can't find this exact example. I assume one of the functions must be 1 and integrate as $\frac{1}{s}$ of course but I have difficulties with the limits of integration and reconstitute the proof.
Also and maybe more importantly, why is integration helping with the error? (just started PID controllers and would appreciate some insights)