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I understand what improper integrals are, but are they really integrals? The semantics are just a bit confusing.
An integral ##\int f(x)\,dx## is a solution to ##F(x)'=f(x)##.I understand what improper integrals are, but are they really integrals? The semantics are just a bit confusing.
Wouldn’t that definition of an integral rule out definite integrals as integrals?An integral ##\int f(x)\,dx## is a solution to ##F(x)'=f(x)##.
As long as there are no boundary conditions to this differential equation, many solutions are possible. Nevertheless, they still have to solve the equation. As such they are a kind of generic solution, the set of possible flows if you like, which we call improper integral (I think; here it is call undetermined). Fixing a boundary condition means to determine a single flow of the vector field, a single solution. So in a way, generic would be the better word, but that's semantics.
Hi,An integral ##\int f(x)\,dx## is a solution to ##F(x)'=f(x)##.
Maybe you're confusing the English terms "indefinite integral" and "improper integral."which we call improper integral (I think; here it is call undetermined).
Yes, I agree, and I agree with your definition of an improper integral.I do believe that is an indefinite integral.
Yes, I did. I couldn't imagine or have forgotten that there is a certain name for integrals with ##\pm \infty## as boundaries. And "undetermined" as literal translation is of course basically the same word as indefinite. Thanks.Maybe you're confusing the English terms "indefinite integral" and "improper integral."
Yes they are integrals, just that they require limits to be solved. I suggest you google Riemann sum. I think the reason they are called improper is because the summation uses limits, I am not sure as to the why they are named so.I understand what improper integrals are, but are they really integrals? The semantics are just a bit confusing.