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Suppose I want to integrate $f(x)+g(x)$. Can this be written as $\int f(x)+g(x)\, dx$ or are brackets necessary, i.e. $\int \left(f(x)+g(x)\right) \,dx$?

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It depends on how fussy or easily confused your readers are. From my point of view, explicit brackets remove possible ambiguity, but there are implicit brackets between $\int$ and $dx$ especially when there is a single integration. –  Henry Apr 15 '12 at 14:38
    
If you believe your readers might have to pause, err on the side of additional parentheses. This isn't like sending a telegraph; nobody meters the use of parentheses in an expression... –  J. M. Apr 15 '12 at 14:48
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I tend to think the brackets are necessary. The $dx$ in an integral is not just a placeholder -- it's a small increment of $x$ that you're multiplying by the function. –  Jim Belk Apr 15 '12 at 16:02
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I fail to see how the version without parenthesis could make sense. If anybody knows references of texts using this convention, please share. –  Did Apr 15 '12 at 18:55

2 Answers 2

up vote -1 down vote accepted

No.

But be careful that in general $$\int f(x) +g(x) d x \neq \int f(x) d x + \int g(x) d x.$$

E.g. consider the integral over $\mathbb{R}$ with $f=1$ and $g(x)=-1$ times the indicator function of $|x| >1$.

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Another one: $\int_0^z \left(\frac1{u}-\frac{\cos\,u}{u}\right)\mathrm du$ is sensible; the split version, not so much... –  J. M. Apr 15 '12 at 14:50
    
@late_learner Is your $g=\chi_{|x|>1}(x)$? –  user21436 Apr 15 '12 at 15:24
    
Depends on your definition;) but probably yes. –  plusepsilon.de Apr 15 '12 at 15:28

This is a conventional exception. If I were going to write $(f(x)+g(x))\,dx$, with no integral sign (e.g. when a differential equation is written as $a(x,y)\,dx+b(x,y)\,dy=0$ and the expression $a(x,y)$ or $b(x,y)$ has several terms) I would not omit the parentheses. Everything within the parentheses is multiplied by $dx$. If $f(x)+g(x)$ is in meters per second and $dx$ is in seconds, then $(f(x)+g(x))\,dx$. However, in something like $\displaystyle\int x^2+3x+10 \, dx$ it is quite conventional to omit delimeters. It is as if the expression $$ \int \cdots\cdots\cdots\cdots dx $$ acts in some way on whatever is written where "$\cdots\cdots\cdots\cdots$" appears.

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