Definition of setting the integration constant equal to zero.

Question

In Calculus books, we often hear of setting the integration constant equal to zero. But what does that actually mean? For example, the set of antiderivatives of $2x$ is $x^2 + C$. If we set the constant equal to zero, we get simply $x^2$. However, I can also write the set of antiderivatives as $x^2 + \pi + C$. In that case, setting the constant equal to zero would give us $x^2 + \pi$. So, my question is, what does setting the constant equal to zero in an antiderivative actually mean?

Answer 1

Mathematically, there isn't a difference - the set of expressions $\{x^2+C:C \in \mathbb{R}\}$ is the same set as the set $\{x^2+\pi+C:C \in \mathbb{R}\}$.

The only difference is that one of those is simpler to read. Written mathematics is a form of communication, and you want to express yourself clearly to the reader. This isn't accomplished just by making statements which start with accepted facts and procedures and lead logically to the conclusion. You also need to minimise the effort the reader puts in to understand your work. This is similar to how you write your answer as $x^2+C,C \in \mathbb{R}$ instead of $x^2+y:y \in \mathbb{R}$: it's mathematically okay, but it differs from convention, and it will leave your reader with a question, "Why did you say it like that?", which means they won't be confident they understand what you are trying to do.

If you say $x^2+\pi+C$, the reader will wonder where the $\pi$ comes from, and why it's important. Maybe it's there for a good reason (e.g. if $x^2+\pi$ was important somewhere else), and in that situation you arguably should leave your answer as $x^2+\pi+C$ instead of $x^2+C$ (I probably still wouldn't if I was writing for other researchers, but I might if I was writing for late high school students). But that means if you leave your answer as $x^2+\pi+C$, the reader will be expecting the $+\pi$ to be important later.

Answer 2

In the rigorous treatment of integration, there is no symbol $\int f(x)dx $ that suggest the degree of freedom in the $+C$, but the idea that still exists. There is only an integrable function $f:[a,b]\to\mathbb{R}$ being integrated over an interval $\int_a^b f$. Notice there is no need for the dummy variable $x,t$ and the "variable of integration" $dx,dt$.

Given a continuous (hence, integrable) function $f:[a,b]\to \mathbb{R}$, we can define the indefinite integral $F$ of $f$ by $$F(x):=\int _a^x f,$$ and this thing has the property that $F'=f$, i.e. $F$ is the antiderivative of $f$. Any antiderivative $G$ of $f$ must satisfy $(G-F)'=G'-F'=f-f=0$, so $G$ and $F$ differ by a constant, write $G=F+C$. Here we see that if we fix a particular antiderivative $F$ to be define like so, then the $+C$ is a fixed number (depend on your $G$), not an arbitrary constant.

I hope this way of putting $+C$ is clear.

Answer 3

Good question! "Setting the integration constant to zero" makes clear sense only in a context where a particular antiderivative has been specified and we want to not append an additive constant to it.

Another example to supplement yours: although $-\arccos x\ne \arcsin x,$ $$-\arccos x+C_1=\int\frac{\mathrm dx}{\sqrt{1-x^2}}=\arcsin x+C_2.$$

Answer 4

As you've pointed out, and as other answers have pointed out, you can redefine your integration constant however you want. So 'setting the constant equal to zero' doesn't have an formal meaning apriori.

Suppose you want to integrate $f(x)$. Then I think what people often mean by 'setting the constant equal to zero' is 'choose an antiderivative which looks most simple to you, and call this one $F(x)$. Write your integral as $\int f(x)dx = F(x)+C$, and then set this $C$ equal to zero'

This definition of 'setting the constant equal to 0' is now subjective and context dependent, because it will be different depending on what functions you find to be most simple. However, it is often a good definition, especially in the case of introductory calculus courses. In this case there is a canonical 'most simple' antiderivative for many integrals considered; eg. when integrating $e^{a x}, x^a, \sin(a x)$ etc, just define $F(x)$ to be the antiderivative with the least number of terms.

If you want to consider antiderivatives for more complex functions, as Korn Kruaykitanon points out, then you may want to consider defining your antiderivative in terms of a definite integral so that $F(x) := \int_b^x f(x')dx'$. You could now for example choose $b=0$ and you'd have a context independent definition of 'setting the constant equal to 0', however in practice $b$ would often still be chosen to simplify a specific problem.

Answer 5

It doesn't really make sense in the general case. What's meant by it is that you assume there is a somehow canonical form of writing the antiderivatives as a sum, and then you remove any constant terms from this sum, thus making it as simple as possible.

In many practical cases there is indeed a “common sense” form where it's obvious which are the constant terms; certainly this is the case for polynomials like your example. But this is not true for arbitrary functions (many people seem to think that a function is a particular algebraic expression, but it's not)). A nice example is $\arccos x = \tfrac\pi2 - \arcsin x$ as in ryang's answer – neither of $\arccos$ and $\arcsin$ is in any meaningful sense preferrable to the other, so it's mostly an arbitrary choice.

It often makes sense to pick the function such that $F(0) = 0$. For polynomials, this corresponds to just omitting the constant term. On the other hand, for exponentials it is not helpful, you would just carry around a $-1$ for no good reason, and if 0 is not even in the domain then this is of course not an option at all.