I am an mechanical engineering student so I'm kind of ashamed to ask this question but I have a weak math background and am digging into some of my knowledge gaps. So my question is where are all of the constants of integration and why are they generally ignored? Is it simply because they are usually treated to be zero? I guess what I'm looking for is an intuition in regards to integration constants...i.e. how do I wrap my mind around why the integration constant is/was ignored, and how can I truly understand what is going on when these constants are ignored. Any insights or conversation is greatly appreciated. Thanks in advance!

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    $\begingroup$ Could you give an example of what you mean (e.g. a problem where constants are being ignored)? $\endgroup$ – user71641 Oct 16 '13 at 2:57
  • $\begingroup$ <EXAMPLE src="db.tt/uFlbKMD3" width="600" height="900"> $\endgroup$ – Michael Travis Oct 16 '13 at 4:32

By the Fundamental Theorem of Calculus, you can think of integration as the reversal of differentiation. Specifically, if you have a continuous function $f:\mathbb{R} \to \mathbb{R}$, and write $$ F(x) = \int_0^x f(t)dt $$ Then $F' = f$.

So now you can ask the question, given $f$ how to I find $F$? One usually finds any function $G$ such that $G' = f$. Now is $G = F$? Almost. The fact is, since $(G-F)' = 0$, they must differ by a constant. Thus, one usually write $$ \int_0^x f(t)dt = G(x) + c $$ where $G$ is any function whose derivative is $f$.

That said, one would ignore the constant only if one is working with

(a) A definite integral (or)

(b) An initial condition (ie. You are given that $F(1) = 3$, say)

  • $\begingroup$ So how do I know if my function is a function of 1,2, or 3 variables. Is it always in relation to the differential. I.E. if I have a dx differential will my constants always be in relation to that differential? And if I have two differentials dxdy how would the constants relate to either x or y (this rare in m. engineering as of now)? Will the constants always have whatever units the integrated function has? For example when we integrate velocity to get position will the constant be a distance measurement? $\endgroup$ – Michael Travis Oct 16 '13 at 4:19

Integration constants are always needed for completeness.

Its absence should not be treated as its value is 0.

Why it is not always shown is a matter of human behavior.

Just imagine that how much effort one has put into the process of successfully "integrated" the function. Our mind is full of joy at that instant. We therefore have the tendency of leaving it behind for the moment. We then add it in when the moment comes or when it is really necessary.

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    $\begingroup$ I don't think "joy" is really a mathematical reason, which is what, I believe, the OP was asking for. $\endgroup$ – Prahlad Vaidyanathan Oct 16 '13 at 3:34
  • $\begingroup$ In my engineering classes it is almost never shown :( I can't even remember the last time I saw a constant of integration in a lecture. $\endgroup$ – Michael Travis Oct 16 '13 at 4:21
  • $\begingroup$ From MT’s post, I think he clearly knows that, after evaluating an indefinite integral, integration constant is needed. “Why are they generally ignored?” verified that. He even queried that ignoring them is “simply because they are usually treated to be zero?” Ignoring them or treating them as 0 is of course mathematically wrong. The main theme is “why the integration constant is/was ignored”. My answer is “joy” – a human behavior makes us temporarily put them aside. For the pt “how can I truly understand what is going on when these constants are ignored" my comment is ‘add (them) in when ...' $\endgroup$ – Mick Oct 16 '13 at 15:38

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