Sorry, just saw that the proof I skipped over because it seemed to assume too much familiarity on my part, gives an explanation of this.
So I'm reading through Ordinary Differential Equations by Tenenbaum and Pollard, page 76, Example 9.5, and I find it bizarre. It says that is exact on which I readily believe. It then says we can integrate each part from to , i.e. . The second integrand baffles me. The book hints that, because we're taking the integral with respect to at the point with fixed, we can therefore eliminate the . But one: Why? Two: If that's so, can't we do the same with the other one and have it be ?