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Is it necessary for someone to write

$$\int (x^2+2x) \,\mathrm{d}x$$

Instead of

$$\int x^2+2x \,\mathrm{d}x$$

With the second one, it's quite obvious which terms we are taking the integral of. Is it still necessary to use brackets in this case?

Skeleton Bow
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    I like to use them. But no, they are not required. –  Nov 23 '16 at 14:58
  • Either one is standard, since you will read the integrand as anything before the $dx$. – Paul Nov 23 '16 at 14:58
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    No. The $\int$ and $dx$ act as their own parentheses. Nevertheless, sometimes it can be aesthetically pleasing to use parentheses. – Eric Auld Nov 23 '16 at 14:58
  • @AdamHughes so by definition the terms(s) integrated are between the $\int$ and the $dx$? (Btw you could post this as an answer) – Skeleton Bow Nov 23 '16 at 15:07
  • @SkeletonBow there you go, answered. – Adam Hughes Nov 23 '16 at 15:09
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    Counterargument: http://math.stackexchange.com/questions/1842911/what-does-it-mean-when-dx-is-put-on-the-start-in-an-integral/1843034#comment3770401_1843034 –  Nov 23 '16 at 15:22
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    Note: not all integrals are single integrals of functions of a single variable. Before too long, you will be doing line integrals like $\int f(x,y)dx + g(x,y)dy$, where the '$\int dx$ are parentheses' logic doesn't hold as tightly. – Jonathan Cast Nov 23 '16 at 16:40
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    If you want to write properly, IMO yes, it's required. – Oriol Nov 23 '16 at 16:48
  • @AdamHughes I know I had chosen your answer as having answered the question, but due to so much back-and-forth debating I decided to let it sit for a while for things to clear. I hope you understand :) – Skeleton Bow Nov 23 '16 at 18:54
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    Note: the highly related question of what is the value of 2+6/2(3). In many cases there is no "right" answer linguistically. This is true in mathematics, at least until some World Governing Body of Mathematics issues a ruling otherwise. – Cort Ammon Nov 23 '16 at 19:21
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    @SkeletonBow I understand from your standpoint of trying to engage in debate. If you take nothing else from mine than this know: as far as notation is concerned you do not need it as the governing wisdom is that this is standard. As always with any notation, there's not really any answer that will satsify anyone completely, because some people want notation to make more psychological sense or relate to something else, but ultimately notation is just convention, so whatever is the convention is "right" and realistically the word "right" is a bad one to use because it different things. – Adam Hughes Nov 23 '16 at 20:07
  • @AdamHughes Your comment quite well sums it up in my opinion. Thank you for your understanding, and your valuable contribution to the discussion! – Skeleton Bow Nov 23 '16 at 20:09
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    @CortAmmon There is a right answer in mathematics for your expression, and it is: "this is a poorly written, ambiguous formula, and has such it has no meaning". It's on the same level as the expression $1+^2)/4$. Just because it's written down doesn't mean that it's meaningful. – Najib Idrissi Nov 24 '16 at 15:03

4 Answers4

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I disagree with the other answers here.

$$\int x^2+2x \,\mathrm{d}x$$

is not correct; the integral needs to be written as

$$\int (x^2+2x) \,\mathrm{d}x$$

instead.

Think of the definite integral, which is really the source of this notation — the definite integral here would be a limit of sums of the form

$$\sum_k (x_k^{\,2}+2x_k) \,\Delta x,$$

not sums of the form

$$\sum_k x_k^{\,2}+2x_k\Delta x.$$

The standard notation works for integrals because you can treat the integral as similar to a summation, and you can treat the part after the integral sign as similar to a product of the integrand and $\mathrm{d}x.$ (Obviously this is just a similarity, not a rigorous definition, but it works in practice.)

Here's an example where it matters: If you want to use a change of variables and apply the substitution rule, you'll get the right answer if you start with

$$\int (x^2+2x) \,\mathrm{d}x$$

and apply the usual laws of algebra, but you will not get the right answer if you start with

$$\int x^2+2x \,\mathrm{d}x$$

instead. (You'll need to add the parentheses back in which should have been there all along.)

For those people who think otherwise, look in published math textbooks or journals and see what kind of usage you find. (If actual usage is different, I would certainly acknowledge that, along with a suggestion then that people should use parentheses when needed to treat this formally as a product of the integrand and $\mathrm{d}x,$ for the reasons I've stated.)

Mitchell Spector
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    In my opinion, this is a well-researched and very reasonable answer. I'm not sure why it was downvoted. I'd like to ask anyone who did to explain why they did so. – Skeleton Bow Nov 23 '16 at 18:24
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    I absolutely agree with you. I see the point of the other opinion; moreover, I even run into integrals without parentheses in some textbooks sometimes — but that always makes me cringe as something wrong. After all, it is a product of $f(x)$ and $dx$, and the rules of orders of operations require the use of parentheses with functions like in the OP's question. – zipirovich Nov 23 '16 at 18:36
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    @SkeletonBow because that's not how it works in real life. Riemann sums and full on integrals have different notations, intuition about making them literally the same notationally just because they are related mathematically is incorrect because it's an apples to oranges comparison. One is on how we write things down, the other is on how it exists in mathematics in the abstract. In particular $dx$ is not literally multiplying the function like $\Delta x$ was before. There are contexts in which that is a multiplication, but not classical calculus. – Adam Hughes Nov 23 '16 at 18:38
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    TL;DR: this is a question on notation for classical calculus, not on the mathematical nature of the objects in question. One can argue this notation should be different, but not that it is different. – Adam Hughes Nov 23 '16 at 18:39
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    My counterargument to those who say that everything between $\int$ and $dx$ is the function would be: what about integrals written as $\int \frac{dx}{x}$? There's nothing in between because $dx$ is not even in the end. This very form makes use of interpreting $dx$ as a multiplicative factor. It's only consistent to always treat it as such and not ignore the basic rules of arithmetic. – zipirovich Nov 23 '16 at 18:40
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    I think you're logically correct, and my preference is to use parentheses. However, a brief skim through Spivak and Apostol, two fairly authoritative calculus texts, yielded mixed results. I found numerous cases where Spivak skips the parentheses when the integrand is relatively simple, whereas Apostol seems to use them consistently even for integrands of the form $x^2 + x$. So it would seem to be at least partially a style decision. –  Nov 23 '16 at 18:40
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    @zipirovich I think some people view the $\int$ and $\mathrm{d}x$ as glorified parentheses, like $\lfloor$ and $\rfloor.$ Doing that, however, deprives you of one of the major advantages of the standard integral notation, as in the change-of-variables example. – Mitchell Spector Nov 23 '16 at 18:41
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    @MitchellSpector: You've just taken it from the tip of my tongue (or my fingertips?)! Kidding, but I was just about to write the same thing: that the change of variable explicitly uses multiplication by $dx$. – zipirovich Nov 23 '16 at 18:44
  • @zipirovich yes, we have some abuses, but that's just a statement that math people do whatever they can psychologically to help themselves. In higher level stuff this is literally a multiplication. The real problem here is that standard conventions are not very clean, and not internally consistent even with the mnemonics. But note that the op never asked about those kinds of ratios either, he asked about whether or not you need parentheses for a distributive property, i.e. multiplication across addition, not multiplication between an atomic function and the $dx$ notation. – Adam Hughes Nov 23 '16 at 18:47
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    @Bungo Does Spivak skip the parentheses where the integrand is a sum or a difference? I'm not claiming that you always need parentheses; you only need them when the priority of operations requires parentheses in order to make the part after the integral sign a formal product of the integrand and $\mathrm{d}x.$ If you could post an example or two from Spivak, I'd be interested. – Mitchell Spector Nov 23 '16 at 18:47
  • @Bungo you are correct, some authors just use more parentheses to avoid the issue entirely. It's still correct to use it without, but personality variations and discomfort with this exact issue (i.e. those who disagree with the common convention) drives people to add more parentheses in. After all, there's nowhere written in the notation rules that adding more notation is illegal, $((2+2))$ is a perfectly well-formed formula, even if I could replace it with the single symbol $4$ or even just $(2+2)$. :) – Adam Hughes Nov 23 '16 at 18:51
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    Here's another example: What does $\int \mathrm{d}x$ mean? It's not the integral of nothing (or of an empty formula); it equals $\int 1 ,\mathrm{d}x,$ in line with the view of the notation as being a formal product. – Mitchell Spector Nov 23 '16 at 18:55
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    @MitchellSpector Here are a couple of examples, from Spivak 4th edition. Page 298, problem 4(i): $\int_0^x 1 + \sin(\sin t)\ dt$. But on the other hand, page 291, $\int (x^3 - x - x^2)\ dx$, so he's not consistent. Spivak being Spivak, most of his integrals are more complicated than these :-) –  Nov 23 '16 at 19:01
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    @Bungo Thanks for the example. Of course, like everything in language, this is a question of what people do in practice, rather than what they arguably should do. If there are many examples of this usage, it's not clear where the line falls between calling these mistakes or idiosyncratic usage and just viewing them as correct usage if it's what people actually do. – Mitchell Spector Nov 23 '16 at 19:20
  • @zipirovich Even with that fraction notation it's well defined, which parts belong to the integrant: Everything between $\int$ and $dx$ and everything that is in the same "fraction" as the $dx$ (save for $dx$ itself) belongs to the integrand. Note that this is not really a fraction you're writing there, since otherwise you could rewrite the integral it as $\frac{1}{x} \cdot\int 1 dx$ – fabian Nov 24 '16 at 10:38
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    While the argument and opinion are certainly acceptable, i have to downvote this because of the final line. I have seen such expressions without parentheses in numerous textbooks and jounal articles. In my experience it is far more common not to include parentheses unless the integrand is unclear, such as in Arghya Chakaborty's example. – Paul Sinclair Nov 24 '16 at 16:23
  • @PaulSinclair The last line was meant literally, as an acknowledgment that this is my opinion, and that it might be worthwhile for someone to survey the literature to judge actual usage; I wasn't presupposing what a large-scale survey would find (but I don't have the time to do one). – Mitchell Spector Nov 24 '16 at 16:29
  • @PaulSinclair I've added to the answer to clarify that. – Mitchell Spector Nov 24 '16 at 16:32
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    That is a pretty weird stance to take. Writing mathematical symbols shouldn't be restricted by their historical significance or the origin of the symbols per se, but rather by utility as a method to convey understanding. When the context is adequately presented and the audience is mature enough, omitting the parenthesis makes it a lot easier to read. – BigbearZzz Nov 29 '16 at 18:07
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By definition of how the notation works when you use the $\displaystyle\int f(x)\,dx $ method, anything between the integral sign and the $dx$ is considered the integrand. If you subscribe to the less-popular school of using the $\displaystyle\int \,dx(f(x))$ then you will need parentheses.

Adam Hughes
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As others have stated it is not required but it is appealing to the eye as: $$\int 2x+3\int x^2+2xy \,\mathrm{dx}\,\mathrm{dy}$$ is hard to read and understand...at least for me. Also as Bye_World stated below, it is hard to determine which variable to integrate in which part.

You can see how messy this double integral is and I might also add that triple integrals are quite common in most of the subjects like 3D mechanics and Differential geometry and many many things.

User Not Found
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  • I think this is the most important point. Using parentheses makes the integral easier to read! – Matthew Leingang Nov 24 '16 at 14:43
  • @MatthewLeingang What do you think about Mitchell's answer. I don't seem to agree with it. He gave the example of substitution but while multiplying he isn't using the distributive property. I think that as long as you use consistent technique you won't run into trouble. What do you think? – User Not Found Nov 25 '16 at 03:15
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The short answer is, yes, you must use the parentheses.

The $dx$ of the integral is not just a delimiter to mark the end of the integral. $dx$ acts somewhat like a variable itself, which the integrand is "multiplied" with.

For example, this:

$$\int \frac{1}{x} \, dx$$

Can equivalently be written as:

$$\int \frac{dx}{x}$$

And you will occasionally see such usage. So long as the expression can be rewritten using the rules of algebra so that the $dx$ is on the right and it is "multiplied" with everything before it, it is a valid integral.

If you write:

$$\int x^2 + 2x \, dx$$

Then you are only "multiplying" the $dx$ with $2x$, not $x^2+2x$, so you do not have a valid integral. People will still know what you mean, but in the same sense people know what you mean if you write "I can haz cheezburger?" — it doesn't make it proper English.

Why treat $dx$ as a variable and do funny things with it? It makes sense when doing differential equations. For example, if you treat $dx$ and $dy$ and variables, then this:

$$\frac{dy}{dx} = xy$$

Can be rewritten using algebra as:

$$\frac{dy}{y} = x \, dx$$

After which you can integrate both sides:

$$\int \frac{dy}{y} = \int x \, dx$$

And you will get the algebra equation $\ln |y| = \frac{1}{2}x^2 + C$. It looks crazy, but this is a standard technique for solving differential equations.

Kef Schecter
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