# Why is there a differential in an indefinite integral?

• Jow
In summary, the notation for antiderivatives as integrals is not arbitrary, but rather a result of the Fundamental Theorem of Calculus. It allows us to link the concepts of differentiation and integration, and provides a clear notation for talking about antiderivatives. While some may argue that the notation can be confusing, it ultimately serves a purpose in communication and understanding of mathematical concepts.
Jow
I understand why a definite integral of the form $^{b}_{a}$∫ƒ(x)dx has the differential dx in it. What I don't understand, and what my teacher hasn't explained is why an indefinite integral (i.e. an antiderivative) requires the differential. Why does ∫ƒ(x)dx require that dx to mean "anti-differentiate"? To put it another way, why is the notation for the antiderivative an integral? It's obviously more than a question of notation, however, as without the differential techniques like u-substitution don't work. I hope this question makes sense, but this is something that has been bothering me for a while now. I figured my teacher would get to it eventually, but he hasn't yet.

Short answer. It tells you what is the variable for integration. Analog is d/dx to tell you the derivative is with respect to x.

Why would techniques like substitution not work, or be dependent on notation? You could replace dx by "Ix" if you wanted and substitution would still work!

$$\int f(u(x))u'(x) Ix = \int f(u) Iu$$

Jow said:
To put it another way, why is the notation for the antiderivative an integral?

It's actually the other way around; we use the notation for integrals to talk about antiderivatives. The Fundamental Theorem of Calculus provides a link between the two concepts.

I am not aware of any notation specifically used to talk about antiderivatives, though I propose we begin using ##\frac{d^{-1}}{dx}f## to denote the family of functions whose derivatives are ##f## (which I'm certain will never happen). Most modern textbooks that I'm familiar with either don't have a notation until integration is introduced or they "cheat" and use the usual notation, ##\int f\ dx##, before students learn about integration with little to no explanation for why that is the chosen notation.

In my opinion, referring to antiderivatives as "indefinite integrals" and denoting them by ##\int f\ dx## is an unnecessary abuse of terminology and notation that serves more to confuse students (and some calculus teachers, I'm sure) than it helps them communicate mathematical ideas.

## 1. Why is there a differential in an indefinite integral?

The differential in an indefinite integral serves as a placeholder for the variable being integrated with respect to. This allows us to express the relationship between the function and its derivative, and ultimately find the antiderivative.

## 2. Can't we just leave out the differential and still get the same result?

No, the differential is an essential part of the indefinite integral notation. Without it, we would not be able to properly represent the antiderivative and its relationship to the original function.

## 3. Why do we use dx as the differential in most cases?

dx is a common choice for a differential in indefinite integrals because it represents a small change in the independent variable x. This aligns with the idea of integration as finding the sum of small changes in a function.

## 4. Is the differential always the same as the variable of integration?

Not necessarily. While in most cases the differential and the variable of integration are the same, there are some cases where they may differ. For example, in cases where the function being integrated is a composite function, the differential may not match the variable of integration.

## 5. Can the differential be any letter or symbol?

Yes, the differential can be any letter or symbol, as long as it is consistent throughout the integration process. Some common choices include dx, du, and dy. However, it is important to note that the differential should not be confused with other variables in the integrand.

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