Bounds of Integration for Nonhomogenous Linear System of Differential Equations

Question

For the method of variation of parameters for a system of linear differential equations, I'm just confused about the integration part. When you integrate from t0 to t, does it matter what t0 you choose as long as it is in the domain? Or are you supposed to choose 0? Because the integral yields different +C answers in both components of the matrix depending on your choice of t0.