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Voltage of  tank circuit is invalid

~~In a [previous](https://electrical.codidact.com/posts/282585) question we found that for a critical damped oscillation:~~
~~![Hi](https://electrical.codidact.com/uploads/YFDDpFztyswF1qeinfqvuZVj)~~
But if we integrate to find the voltage through the capacitor:
~~![hI](https://electrical.codidact.com/uploads/Yx3t6FRnMKZTqBAPGycqk2ra)~~
~~the integral is always a positive value which doesnt make any sense since the capacitor is being discharged.~~
~~What am I doing wrong here?Help appreciated!~~

In a [previous](https://electrical.codidact.com/posts/282585) question we found that for a critically damped oscillation:
$$
I(t) = 2 t e^{-0.5 t}
$$
But if we integrate to find the voltage through the capacitor:
$$
V_{CO} + \frac{1}{C} \cdot \int_0^t{I(t)}dt
$$
the integral is always a positive value which doesn't make any sense since the capacitor is being discharged.
What am I doing wrong here? Help appreciated!

voltage