The difficulty here stems from the fact of dealing with an improper integral, in this case integrating over an unbounded region where the basic function never converges. Of course there are techniques for evaluating improper integrals but they all require convergence. Nevertheless, integrating from t=-infinity is theoretically correct but not applicable to a real physical system, i.e. the capacitor. If instead of minus infinity you begin the integration no further back than the creation of the fabric of space-time some 13.7 billion years ago, then evaluation of the definite integral will be possible.