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It's curious that you can have different constants of integration on intervals. E.g. if $$f(x) = \left\{ \begin{array}{llr} \frac{-1}{x}+a, & x>0\\ \frac{-1}{x}+b, & x<0\\ \end{array} \right. $$ then $$ \frac{\mathrm{d}}{\mathrm{d}x}f(x) = \frac{1}{x^2}.$$ Are there situations/applications where different constants of integration on different intervals are needed/required?

I'm interested in this issue because many textbooks gloss over this issue. They prove results on an interval. Many of these books then have exercises like "Find $\int \frac{1}{x^2}\mathrm{d}x$", but don't mention the potential for different constants of integration. (Some books do mention this of course, notably Stewart's Calculus.) I'm trying to decide how much this issue "matters". If there is a wide range of compelling applications in which different constants of integration occurs, it suggests we should pay more attention.

I've asked around colleagues and students (E.g. I have about 850 students in my current calculus class) and very few people are aware of this, but they do think it's curious when pointed out explicitly....

Chris Sangwin
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    One way in which integration arises "naturally" is when solving differential equations in applied mathematics. In my experience with applied mathematics (which is not extensive, since my PhD is in number theory), it seems to be the case that, when the solution of the ODE blows up, that's where the validity of the model ends. In such cases, the issue of multiple constants of integration doesn't come up, since you're only considering the function that is to be integrated on a single interval where it is well-defined. More generally, an antiderivative with multiple different choices of... – R.P. Oct 07 '24 at 16:18
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    ...constants of integration seems to be a very artificial construct. But I am not sure if there are not some applications of this idea. – R.P. Oct 07 '24 at 16:20
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    Hi, welcome to Math SE. To take a physics example, look at gravitational potential inside & outside a sphere of uniform density. – J.G. Oct 07 '24 at 17:36
  • Just to point out that the integrand doesn't even need to be explicitly defined on intervals for its indefinite integral to require independent arbitrary constants: $\dfrac1x$. – ryang Oct 08 '24 at 12:12
  • @ryang isn't that already the case for the example in the question? – ronno Oct 08 '24 at 12:21
  • If a function has an antiderivative at every point on an interval, then any two antiderivatives on that interval will differ by a single "constant of integration". The situation here occurs because $f$ is not differentiable at $0$, which means the domain of $f'$ consists of two intervals. It is this disconnectedness that allows the constant of integration to change between the intervals. A function only has to be "locally constant" to have $0$ derivative. An interuption of domain allows that local constant to change. – Paul Sinclair Oct 08 '24 at 14:22
  • Yes @PaulSinclair that's the whole point of the example! It's an issue which is glossed over in many calculus books. That's why I'm interested in finding out whether it's ever useful in applications to have differing constants on different intervals. – Chris Sangwin Oct 08 '24 at 18:13
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    J.G. gave you an excellent example. Similar examples abound. When you model physical objects, they usually have surfaces, which we model as sharp discontinuities between what is inside the object and what is outside. Any such change expresses itself as discontinuities in functions, and their derivatives. Some of them satisfy the same differential equations inside and outside, but the discontinuity can be considered to be exactly a different constant of integration. Usually, though, we treat the two sides separately, so we don't notice it. – Paul Sinclair Oct 09 '24 at 00:57
  • @ronno Oh yes, thanks; I'd skimmed and thought the OP's $f$ to be the integrand. – ryang Oct 09 '24 at 11:49

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