If I'm not mistaken, the TI derivative/anti-derivatives solving methods rely on approximations when the answer can't be obtained by the logic programmed into these methods.
In other words, where there's insufficient logic programmed, the TI cranks out an approximation, then based on user input, makes an extremely educated guess as to what the "exact" answer is.
I would think that guess is better than nearly all humans could make, and was necessary because nearly all humans can't program sufficient logic to come up with the dx or anti-dx for every possible expression.
i can tell you from firsthand experience the TI symbolic integration and differentiation capabilities are extremely limited. a lot of times the answer it would offer up was just plain wrong, or it would just throw an error.
If it were not limited, it would not require the user to input a "guess."
are you talking about the ODE solver? integration doesn't require guesses.