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?
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asked Nov 23, 2016 at 14:56
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7$\begingroup$ No. The $\int$ and $dx$ act as their own parentheses. Nevertheless, sometimes it can be aesthetically pleasing to use parentheses. $\endgroup$– Eric AuldNov 23, 2016 at 14:58
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3$\begingroup$ Counterargument: math.stackexchange.com/questions/1842911/… $\endgroup$– user856Nov 23, 2016 at 15:22
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7$\begingroup$ 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. $\endgroup$– Jonathan CastNov 23, 2016 at 16:40
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3$\begingroup$ If you want to write properly, IMO yes, it's required. $\endgroup$– OriolNov 23, 2016 at 16:48
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3$\begingroup$ 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. $\endgroup$– Cort AmmonNov 23, 2016 at 19:21
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4 Answers
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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.)
answered Nov 23, 2016 at 18:00
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5$\begingroup$ 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. $\endgroup$ Nov 23, 2016 at 18:36
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13$\begingroup$ @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. $\endgroup$ Nov 23, 2016 at 18:38
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8$\begingroup$ 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. $\endgroup$ Nov 23, 2016 at 18:39
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6$\begingroup$ 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. $\endgroup$ Nov 23, 2016 at 18:40
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4$\begingroup$ 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. $\endgroup$ Nov 23, 2016 at 18:55
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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.
answered Nov 23, 2016 at 15:09
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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.
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$\begingroup$ I think this is the most important point. Using parentheses makes the integral easier to read! $\endgroup$ Nov 24, 2016 at 14:43
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$\begingroup$ @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? $\endgroup$ Nov 25, 2016 at 3:15
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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.
