24 Oct '19 07:31>7 edits
Just checking; is the following always true as a rule?
∫ f(x) dx = ∫[0, x] f(x) dx + C ⇐ ∫[0, x] f(x) dx ∈ ℝ
where C is an arbitrary constant.
(The implied precondition from "⇐ ∫[0, x] f(x) dx ∈ ℝ" part above is required because I know "∫ f(x) dx = ∫[0, x] f(x) dx + C" doesn't work for such cases as f(x) = 1/x etc. because the ∫[0, x] f(x) dx integral for such cases doesn't converge to a real number)
I think it surely must be but I only ask because I would like to put it in my book I am writing but I have thoroughly search the net for it and strangely cannot find this seemingly obvious rule (or implied equivalent) mentioned ANYWHERE over the net even though it seems obvious to me and this made me wonder why; Is something wrong with it? Is it always technically correct but nevertheless for some reason considered bad practice to think in terms of it?
∫ f(x) dx = ∫[0, x] f(x) dx + C ⇐ ∫[0, x] f(x) dx ∈ ℝ
where C is an arbitrary constant.
(The implied precondition from "⇐ ∫[0, x] f(x) dx ∈ ℝ" part above is required because I know "∫ f(x) dx = ∫[0, x] f(x) dx + C" doesn't work for such cases as f(x) = 1/x etc. because the ∫[0, x] f(x) dx integral for such cases doesn't converge to a real number)
I think it surely must be but I only ask because I would like to put it in my book I am writing but I have thoroughly search the net for it and strangely cannot find this seemingly obvious rule (or implied equivalent) mentioned ANYWHERE over the net even though it seems obvious to me and this made me wonder why; Is something wrong with it? Is it always technically correct but nevertheless for some reason considered bad practice to think in terms of it?