Parallel capacitors with initial voltage
If you have two capacitors connected in parallel, each with initial voltage Vi, then the initial voltage of the pair will also be Vi. Two elements in parallel will always have the same voltage across them.

As the current is going to charge the total capacity of 2C, the voltage with respect to time will be \$V(t) = V_i +  \frac{1}{2C}\int{I(t)}dt\$, not 2Vi + ... as you've written.