Voltage across a capacitor variable of integration
This is really just a math question.

When you take the integral \$ \frac{1}{C}\int_{t_0}^t I(\tau)d\tau \$, you get \$ \frac{Q(t)}{C} - \frac{Q(t_0)}{C} \$ = \$ V(t) - V(t_0)\$. This is bad, because we just want \$ V(t) \$; the simple way of fixing this is to add \$V(t_0)\$ to cancel out the term in the integral.

We definitely can't just write \$ \frac{1}{C} \int I(t) \$ - what are you going to do with the constant of integration?