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They are just deriving the equation from a definite integral. In both situations the parameters are 0 to I. In the second case they are just using the time it takes to discharge from I which is theoretically infinity (notice the decay equation added in ) .
The derivation of the energy stored in an inductor does start out as an integral with respect to time. It then uses the relations among differentials to change variables.
(dU/dt)dt = dU and (dI/dt)dt = dI
That's just basic integral calculus. The change of variables permits the integration to be performed without knowing the specific form of I as a function of t and shows that the result is valid for all functions of t.
The second result is the calculation of the energy dissipated in a resistor during the exponential decay in an RL circuit. It's an independent calculation that demonstrates conservation of energy. The energy dissipated in the resistor during the decay is exactly equal to the energy that was stored in the inductor.