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. $\int \left(f(x)+g(x)\right) \,dx$?
Necessity of Brackets for Integration
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3It 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. – 2012-04-15
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0If 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... – 2012-04-15
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4I 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. – 2012-04-15
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3I fail to see how the version without parenthesis could make sense. If anybody knows references of texts using this convention, please share. – 2012-04-15
2 Answers
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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0Another one: $\int_0^z \left(\frac1{u}-\frac{\cos\,u}{u}\right)\mathrm du$ is sensible; the split version, not so much... – 2012-04-15
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0@late_learner Is your $g=\chi_{|x|>1}(x)$? – 2012-04-15
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0Depends on your definition;) but probably yes. – 2012-04-15
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.