Deriving the Integral Voltage-Current Relationship of a Capacitor
In the interest of having an actual answer posted after working this out in comments:

This is not a universal rule. It only works for certain types of current waveforms, specifically when

$$\lim_{t_0\to -\infty}\int_{t_0}^{t_1} i(t) {\rm d}t$$ is well defined for some finite \$t_1\$.

It is reasonable to assume the "initial" voltage as \$t\to-\infty\$ is 0 because the leakage current in real capacitors means that any charge that was present a sufficiently long time ago will have leaked away and not affect the behavior circuit in the time range we're actually interested in.