sean-fitzpatrick
commented
3 days ago I'm fine with this change. Since we define the indefinite integral as a set, the notation is a little awkward, but in practice I think everyone thinks of the indefinite integral as a representative of this set, with the value of C left unspoken.
Probably the best option (mathematically, not pedagocially) is to think of this set as an equivalence class: we have many examples throughout mathematics of operations on equivalence classes.
Possible correction to antidiff. property.
Theorem 5.1.6 states \int c \cdot f(x) \, dx = c \cdot \int f(x) \, dx. While this is identical to language in other texts, I got a suggestion that we should add "(c \neq 0)" as a condition.
The reason (which you can figure on your own) is that if c=0, integrating 0 leads to a +C, whereas integrating first then multiplying by 0 means you just get 0.
I'm wondering what others think of this suggestion. I'm lean toward making the change. Yes, I think they have a point. I also think the expression "c \cdot \int f(x)\,dx" is also playing a bit loose with set notation, and this may be an unimportant nitpick. But if it gets students thinking about what's going on, it may create a bit of a learning moment.
Interested in any discussion before someone merges.